Theorem rexeqbii 2646

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 2417	. . 3 ⊢ (x ∈ A ↔ x ∈ B)
3		raleqbii.2	. . 3 ⊢ (ψ ↔ χ)
4	2, 3	anbi12i 678	. 2 ⊢ ((x ∈ A ∧ ψ) ↔ (x ∈ B ∧ χ))
5	4	rexbii2 2644	1 ⊢ (∃x ∈ A ψ ↔ ∃x ∈ B χ)

This proof depends on syntax axioms:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-rex 2621

This theorem is used by: (None)