Theorem rexcom 2773

Description: Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

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

Distinct variable groups:   x,y   x,B   y,A

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

Proof of Theorem rexcom

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲyA
2		nfcv 2490	. 2 ⊢ ℲxB
3	1, 2	rexcomf 2771	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)

Syntax hints:   ↔ wb 176  ∃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:  rexcom13  2774  rexcom4  2879  iuncom  3976  addccom  4407  ltfinex  4465  nnpw1ex  4485  evenodddisjlem1  4516  phialllem1  4617  xpiundi  4818  addcfnex  5825  crossex  5851  mucex  6134  ncspw1eu  6160  addlec  6209  nc0le1  6217  dflec3  6222