Theorem rexeqbii 3336

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 2855	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rexeqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 640	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	rexbii2 3108	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ∃wrex 3089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2755  df-clel 2838  df-rex 3090

This theorem is used by:  1arithidom  34122  bnj882  35609  satfbrsuc  36252  iuneq12i  37111  setc1onsubc  50879