Theorem rexcom 2772

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 2489	. 2 ⊢ ℲyA
2		nfcv 2489	. 2 ⊢ ℲxB
3	1, 2	rexcomf 2770	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)

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-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 2478  df-rex 2620

This theorem is referenced by:  rexcom13  2773  rexcom4  2878  iuncom  3975  addccom  4406  ltfinex  4464  nnpw1ex  4484  evenodddisjlem1  4515  phialllem1  4616  xpiundi  4817  addcfnex  5824  crossex  5850  mucex  6133  ncspw1eu  6159  addlec  6208  nc0le1  6216  dflec3  6221