Theorem rexeqbii 3344

Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
rexeqbii.1	⊢ 𝐴 = 𝐵
rexeqbii.2	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
rexeqbii	⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)

Proof of Theorem rexeqbii

Step	Hyp	Ref	Expression
1		rexeqbii.1	. . . 4 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2832	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rexeqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 628	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	rexbii2 3089	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2107  ∃wrex 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2728  df-clel 2815  df-rex 3070

This theorem is referenced by:  1arithidom  33566  bnj882  34941  satfbrsuc  35372  iuneq12i  36197