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Mirrors > Home > MPE Home > Th. List > rexeq | Structured version Visualization version GIF version |
Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
rexeq | ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexeqf 3112 | 1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 |
This theorem is referenced by: rexeqi 3120 rexeqdv 3122 rexeqbi1dv 3124 unieq 4380 exss 4858 qseq1 7683 1sdom 8048 pssnn 8063 indexfi 8157 supeq1 8234 bnd2 8639 dfac2 8836 cflem 8951 cflecard 8958 cfeq0 8961 cfsuc 8962 cfflb 8964 cofsmo 8974 elwina 9387 eltskg 9451 rankcf 9478 elnp 9688 elnpi 9689 genpv 9700 xrsupsslem 12009 xrinfmsslem 12010 xrsupss 12011 xrinfmss 12012 hashge2el2difr 13118 isdrs 16757 isipodrs 16984 neifval 20713 ishaus 20936 2ndc1stc 21064 1stcrest 21066 lly1stc 21109 isref 21122 islocfin 21130 tx1stc 21263 isust 21817 iscfilu 21902 met1stc 22136 iscfil 22871 isgrpo 26735 chne0 27737 pstmfval 29267 dya2iocuni 29672 nobndlem2 31092 nobndlem8 31098 altxpeq1 31250 altxpeq2 31251 elhf2 31452 bj-sngleq 32148 cover2g 32679 indexdom 32699 istotbnd 32738 pmapglb2xN 34076 paddval 34102 elpadd0 34113 diophrex 36357 hbtlem1 36712 hbtlem7 36714 |
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