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 2861	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rexeqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 639	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	rexbii2 3114	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-clel 2844  df-rex 3096

This theorem is referenced by:  1arithidom  33772  bnj882  35259  satfbrsuc  35757  iuneq12i  36596  setc1onsubc  50265