Theorem rexeqbii 2337

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
raleqbii.1	⊢ 𝐴 = 𝐵
raleqbii.2	⊢ (𝜓 ↔ 𝜒)

Assertion

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

Proof of Theorem rexeqbii

Step	Hyp	Ref	Expression
1		raleqbii.1	. . . 4 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2104	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		raleqbii.2	. . 3 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 433	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	rexbii2 2335	1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-rex 2312

This theorem is referenced by: (None)