Theorem rexcom 2672

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 2350	. 2 |- F/_ y A
2		nfcv 2350	. 2 |- F/_ x B
3	1, 2	rexcomf 2670	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 2487

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492

This theorem is referenced by:  rexcom13  2674  rexcom4  2800  iuncom  3947  xpiundi  4751  addcomprg  7726  mulcomprg  7728  ltexprlemm  7748  caucvgprprlemexbt  7854  suplocexprlemml  7864  suplocexprlemmu  7866  qmulz  9779  elpq  9805  caubnd2  11543  sqrt2irr  12599  pythagtriplem19  12720