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Mirrors > Home > ILE Home > Th. List > rexeqbidv | GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Ref | Expression |
---|---|
raleqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexeqbidv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | rexeqdv 2556 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
3 | raleqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 3 | rexbidv 2369 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | bitrd 186 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 |
This theorem is referenced by: supeq123d 6404 |
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