Theorem 2rexbii 2641

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 2639	. 2 ⊢ (∃y ∈ B φ ↔ ∃y ∈ B ψ)
3	2	rexbii 2639	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ)

Syntax hints:   ↔ wb 176  ∃wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-rex 2620

This theorem is referenced by:  3reeanv  2779  addccom  4406  addcass  4415  ncfinraise  4481  ncfinlower  4483  nnpweq  4523  dfxp2  5113  peano4nc  6150  sbth  6206