Theorem bi2rexa 1230

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

Hypothesis

Ref	Expression
bi2rexa.1	⊢ ((x ∈ A ∧ y ∈ B) → (φ ↔ ψ))

Assertion

Ref	Expression
bi2rexa	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Distinct variable group(s):   x,y   y,A

Proof of Theorem bi2rexa

Step	Hyp	Ref	Expression
1		bi2rexa.1	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → (φ ↔ ψ))
2	1	birexdva 1216	. 2 ⊢ (x ∈ A → (∃y ∈ B φ ↔ ∃y ∈ B ψ))
3	2	birexa 1229	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  elrnoprab 3054  sqr2irr 4782  mdsymlem8 5783

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  ax-17 925

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

metamath.org