Theorem rexcom 2695

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 2372	. 2 |- F/_ y A
2		nfcv 2372	. 2 |- F/_ x B
3	1, 2	rexcomf 2693	1 |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )

Syntax hints:   <-> wb 105   E.wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514

This theorem is referenced by:  rexcom13  2697  rexcom4  2823  iuncom  3970  xpiundi  4776  addcomprg  7761  mulcomprg  7763  ltexprlemm  7783  caucvgprprlemexbt  7889  suplocexprlemml  7899  suplocexprlemmu  7901  qmulz  9814  elpq  9840  caubnd2  11623  sqrt2irr  12679  pythagtriplem19  12800