Theorem rexcom 2598

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 2282	. 2 |- F/_ y A
2		nfcv 2282	. 2 |- F/_ x B
3	1, 2	rexcomf 2596	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 2418

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423

This theorem is referenced by:  rexcom13  2599  rexcom4  2712  iuncom  3827  xpiundi  4605  addcomprg  7410  mulcomprg  7412  ltexprlemm  7432  caucvgprprlemexbt  7538  suplocexprlemml  7548  suplocexprlemmu  7550  qmulz  9442  elpq  9467  caubnd2  10921  sqrt2irr  11876