Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2822 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | rexeqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | anbi12i 630 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) |
5 | 4 | rexbii2 3158 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2728 df-clel 2809 df-rex 3057 |
This theorem is referenced by: bnj882 32573 satfbrsuc 32995 |
Copyright terms: Public domain | W3C validator |