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Mirrors > Home > ILE Home > Th. List > rexeqdv | GIF version |
Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
Ref | Expression |
---|---|
raleq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
rexeqdv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | rexeq 2674 | . 2 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 |
This theorem is referenced by: rexeqbidv 2686 rexeqbidva 2688 fnunirn 5771 cbvexfo 5790 fival 6972 nninfwlpoimlemg 7176 nninfwlpoimlemginf 7177 nninfwlpoim 7179 genipv 7511 exfzdc 10243 zproddc 11590 infssuzex 11953 nninfdcex 11957 ennnfonelemrnh 12420 grppropd 12899 dvdsrpropdg 13322 cnpfval 13835 |
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