Theorem rexcom 2670

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 2348	. 2 |- F/_ y A
2		nfcv 2348	. 2 |- F/_ x B
3	1, 2	rexcomf 2668	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 2485

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490

This theorem is referenced by:  rexcom13  2672  rexcom4  2795  iuncom  3933  xpiundi  4733  addcomprg  7691  mulcomprg  7693  ltexprlemm  7713  caucvgprprlemexbt  7819  suplocexprlemml  7829  suplocexprlemmu  7831  qmulz  9744  elpq  9770  caubnd2  11428  sqrt2irr  12484  pythagtriplem19  12605