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Mirrors > Home > MPE Home > Th. List > rexeqbid | Structured version Visualization version GIF version |
Description: Equality deduction for restricted existential quantifier. See rexeqbidv 3344 for a version based on fewer axioms. (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 3351 | . . 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 3272 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
9 | 5, 8 | bitrd 279 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 Ⅎwnf 1786 Ⅎwnfc 2884 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 |
This theorem is referenced by: iuneq12df 5022 |
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