Theorem rexcom 2634

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 2312	. 2 |- F/_ y A
2		nfcv 2312	. 2 |- F/_ x B
3	1, 2	rexcomf 2632	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 2449

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  rexcom13  2635  rexcom4  2753  iuncom  3879  xpiundi  4669  addcomprg  7540  mulcomprg  7542  ltexprlemm  7562  caucvgprprlemexbt  7668  suplocexprlemml  7678  suplocexprlemmu  7680  qmulz  9582  elpq  9607  caubnd2  11081  sqrt2irr  12116  pythagtriplem19  12236