Sorry, no online dictionaries contain the exact phrase first-order logic with equality.

Reverse dictionary results:
1. leading
2. premier
3. prime
4. prominent
5. subordinate
6. egalitarian
7. egalitarianism
8. emulate
9. equalled
10. equals
11. equation
12. equilibrium
13. equity
14. even
15. evens
16. fairness
17. feminism
18. isodynamic
19. isometric
20. isothermal
21. king
22. match
23. matched
24. non-discrimination
25. square
26. symmetry
27. analogy
28. associate
29. parity
30. democracy
31. adjunct
32. complement
33. determination
34. dichotomy
35. dilemma
36. equivalent
37. illogic
38. irrational
39. middle
40. qualification

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You can look up the words in the phrase individually using these links:   first-order ?   logic ?   with ?   equality ?
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