Theorem rexcom 2593

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 2279	. 2 |- F/_ y A
2		nfcv 2279	. 2 |- F/_ x B
3	1, 2	rexcomf 2591	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 2415

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420

This theorem is referenced by:  rexcom13  2594  rexcom4  2704  iuncom  3814  xpiundi  4592  addcomprg  7379  mulcomprg  7381  ltexprlemm  7401  caucvgprprlemexbt  7507  suplocexprlemml  7517  suplocexprlemmu  7519  qmulz  9408  caubnd2  10882  sqrt2irr  11829