Theorem r2ex 2653

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)

Assertion

Ref	Expression
r2ex	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ))

Distinct variable groups:   x,y   y,A

Allowed substitution hints:   φ(x,y)   A(x)   B(x,y)

Proof of Theorem r2ex

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲyA
2	1	r2exf 2651	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃wrex 2616

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by:  reean  2778  elxpk2  4198  evenfinex  4504  oddfinex  4505  rnoprab2  5578  rnmpt2  5718  lecex  6116  mucnc  6132