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Mirrors > Home > MPE Home > Th. List > rexeqbii | Structured version Visualization version GIF version |
Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
rexeqbii.1 | ⊢ 𝐴 = 𝐵 |
rexeqbii.2 | ⊢ (𝜓 ↔ 𝜒) |
Ref | Expression |
---|---|
rexeqbii | ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeqbii.1 | . . . 4 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2898 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | rexeqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | anbi12i 620 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) |
5 | 4 | rexbii2 3249 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1879 df-cleq 2818 df-clel 2821 df-rex 3123 |
This theorem is referenced by: bnj882 31538 |
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