Theorem bi2rex 1226

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

Hypothesis

Ref	Expression
biral.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
bi2rex	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Proof of Theorem bi2rex

Step	Hyp	Ref	Expression
1		biral.1	. . 3 ⊢ (φ ↔ ψ)
2	1	birex 1224	. 2 ⊢ (∃y ∈ B φ ↔ ∃y ∈ B ψ)
3	2	birex 1224	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   ↔ wb 127  ∃wrex 1202

This theorem is referenced by:  elrnoprab 3054  unxpdomlem 3649

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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