Theorem rexcom4a 3226

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 3225	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. x E. y e. A ( ph /\ ps ) )
2		19.42v 1918	. . 3 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
3	2	rexbii 3041	. 2 |- ( E. y e. A E. x ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )
4	1, 3	bitr3i 266	1 |- ( E. x E. y e. A ( ph /\ ps ) <-> E. y e. A ( ph /\ E. x ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  rexcom4b  3227