Theorem 2rexbii 2327

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)

Hypothesis

Ref	Expression
ralbii.1	⊢ (φ ↔ ψ)

Assertion

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

Proof of Theorem 2rexbii

Step	Hyp	Ref	Expression
1		ralbii.1	. . 3 ⊢ (φ ↔ ψ)
2	1	rexbii 2325	. 2 ⊢ (∃y ∈ B φ ↔ ∃y ∈ B ψ)
3	2	rexbii 2325	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   ↔ wb 98  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-rex 2306

This theorem is referenced by:  3reeanv  2474  4fvwrd4  8767