Theorem rexcom 2707

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 2384	. 2 |- F/_ y A
2		nfcv 2384	. 2 |- F/_ x B
3	1, 2	rexcomf 2705	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 2521

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526

This theorem is referenced by:  rexcom13  2709  rexcom4  2837  iuncom  3997  xpiundi  4808  addcomprg  7893  mulcomprg  7895  ltexprlemm  7915  caucvgprprlemexbt  8021  suplocexprlemml  8031  suplocexprlemmu  8033  qmulz  9955  elpq  9981  caubnd2  11802  sqrt2irr  12859  pythagtriplem19  12980