Theorem rexcom4a 2705

Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Assertion

Ref	Expression
rexcom4a	|- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

Distinct variable groups:   x, A   x, y   ph, x

Allowed substitution hints:   ph( y)   ps( x, y)   A( y)

Proof of Theorem rexcom4a

Step	Hyp	Ref	Expression
1		rexcom4 2704	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. x E. y e. A ( ph /\ ps ) )
2		19.42v 1878	. . 3 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
3	2	rexbii 2440	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
4	1, 3	bitr3i 185	1 |- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1468   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-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683

This theorem is referenced by:  rexcom4b  2706