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Mirrors > Home > ILE Home > Th. List > rexeq | 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 2312 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2312 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexeqf 2662 | 1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 ∃wrex 2449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 |
This theorem is referenced by: rexeqi 2670 rexeqdv 2672 rexeqbi1dv 2674 unieq 3803 bnd2 4157 exss 4210 qseq1 6557 finexdc 6876 supeq1 6959 isomni 7108 ismkv 7125 sup3exmid 8860 exmidunben 12368 neifval 12893 cnprcl2k 12959 bj-nn0sucALT 13973 strcoll2 13978 strcollnft 13979 strcollnfALT 13981 sscoll2 13983 |
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