Theorem rexcom 2532

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 2229	. 2 |- F/_ y A
2		nfcv 2229	. 2 |- F/_ x B
3	1, 2	rexcomf 2530	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 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366

This theorem is referenced by:  rexcom13  2533  rexcom4  2643  iuncom  3742  xpiundi  4509  addcomprg  7198  mulcomprg  7200  ltexprlemm  7220  caucvgprprlemexbt  7326  qmulz  9169  caubnd2  10611  sqrt2irr  11480