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Mirrors > Home > MPE Home > Th. List > rexeqbid | Structured version Visualization version GIF version |
Description: Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
raleqbid.0 | ⊢ Ⅎ𝑥𝜑 |
raleqbid.1 | ⊢ Ⅎ𝑥𝐴 |
raleqbid.2 | ⊢ Ⅎ𝑥𝐵 |
raleqbid.3 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbid.4 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexeqbid | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbid.3 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | raleqbid.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | raleqbid.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | rexeqf 3112 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
6 | raleqbid.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
7 | raleqbid.4 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
8 | 6, 7 | rexbid 3033 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
9 | 5, 8 | bitrd 267 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 Ⅎwnf 1699 Ⅎwnfc 2738 ∃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: iuneq12df 4480 |
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