Theorem 2rexbiia 1678

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 1663	. 2 ⊢ (x ∈ A → (∃y ∈ B φ ↔ ∃y ∈ B ψ))
3	2	rexbiia 1677	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ∈ wcel 960  ∃wrex 1649

This theorem is referenced by:  sqr2irr 6730  mdsymlem8 10332

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-rex 1653

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