Theorem rexeqbii 3202

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 2842	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rexeqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 612	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	rexbii2 3187	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-clel 2767  df-rex 3067

This theorem is referenced by:  bnj882  31334