Theorem rexcom 2629

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

Assertion

Ref	Expression
rexcom	|- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )

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

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

Proof of Theorem rexcom

Step	Hyp	Ref	Expression
1		nfcv 2307	. 2 |- F/_ y A
2		nfcv 2307	. 2 |- F/_ x B
3	1, 2	rexcomf 2627	1 |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )

Syntax hints:   <-> wb 104   E.wrex 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449

This theorem is referenced by:  rexcom13  2630  rexcom4  2748  iuncom  3871  xpiundi  4661  addcomprg  7515  mulcomprg  7517  ltexprlemm  7537  caucvgprprlemexbt  7643  suplocexprlemml  7653  suplocexprlemmu  7655  qmulz  9557  elpq  9582  caubnd2  11055  sqrt2irr  12090  pythagtriplem19  12210