Theorem 2rexbiia 1667

Description: Inference adding 2 restricted existential quantifiers to both sides of an equivalence.

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

Proof of Theorem 2rexbiia

Step	Hyp	Ref	Expression
1		2rexbiia.1	. . 3 ⊢ ((x ∈ A ⋀ y ∈ B) → (φ ↔ ψ))
2	1	rexbidva 1652	. 2 ⊢ (x ∈ A → (∃y ∈ B φ ↔ ∃y ∈ B ψ))
3	2	rexbiia 1666	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 955  ∃wrex 1638

This theorem is referenced by:  sqr2irr 6659  mdsymlem8 10245

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-rex 1642

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