Theorem rexcom4a 2578

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 2577	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. x E. y e. A ( ph /\ ps ) )
2		19.42v 1786	. . 3 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
3	2	rexbii 2331	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
4	1, 3	bitr3i 175	1 |- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559

This theorem is referenced by:  rexcom4b  2579