Theorem rexcom4a 2824

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 2823	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. x E. y e. A ( ph /\ ps ) )
2		19.42v 1953	. . 3 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
3	2	rexbii 2537	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
4	1, 3	bitr3i 186	1 |- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   E.wrex 2509

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801

This theorem is referenced by:  rexcom4b  2825