Theorem rexbiia 2648

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)

Hypothesis

Ref	Expression
ralbiia.1	⊢ (x ∈ A → (φ ↔ ψ))

Assertion

Ref	Expression
rexbiia	⊢ (∃x ∈ A φ ↔ ∃x ∈ A ψ)

Proof of Theorem rexbiia

Step	Hyp	Ref	Expression
1		ralbiia.1	. . 3 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.32i 618	. 2 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ))
3	2	rexbii2 2644	1 ⊢ (∃x ∈ A φ ↔ ∃x ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-rex 2621

This theorem is referenced by:  2rexbiia  2649  ceqsrexbv  2974  reu8  3033  phialllem1  4617  finnc  6244  nchoicelem11  6300  nchoicelem16  6305