Theorem 2rexbiia 2649

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Hypothesis

Ref	Expression
2rexbiia.1	⊢ ((x ∈ A ∧ y ∈ B) → (φ ↔ ψ))

Assertion

Ref	Expression
2rexbiia	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Distinct variable groups:   x,y   y,A

Allowed substitution hints:   φ(x,y)   ψ(x,y)   A(x)   B(x,y)

Proof of Theorem 2rexbiia

Step	Hyp	Ref	Expression
1		2rexbiia.1	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → (φ ↔ ψ))
2	1	rexbidva 2632	. 2 ⊢ (x ∈ A → (∃y ∈ B φ ↔ ∃y ∈ B ψ))
3	2	rexbiia 2648	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ 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  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is referenced by: (None)