Theorem 2rexbii 2516

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 2514	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)
3	2	rexbii 2514	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 105  ∃wrex 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-rex 2491

This theorem is referenced by:  3reeanv  2678  4fvwrd4  10275  prodmodc  11939  pythagtriplem2  12639  pythagtrip  12656