Theorem rexcom4b 2714

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

Hypothesis

Ref	Expression
rexcom4b.1	|- B e. _V

Assertion

Ref	Expression
rexcom4b	|- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

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

Allowed substitution hints:   ph( y)   A( y)   B( y)

Proof of Theorem rexcom4b

Step	Hyp	Ref	Expression
1		rexcom4a 2713	. 2 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ( ph /\ E. x x = B ) )
2		rexcom4b.1	. . . . 5 |- B e. _V
3	2	isseti 2697	. . . 4 |- E. x x = B
4	3	biantru 300	. . 3 |- ( ph <-> ( ph /\ E. x x = B ) )
5	4	rexbii 2445	. 2 |- ( E. y e. A ph <-> E. y e. A ( ph /\ E. x x = B ) )
6	1, 5	bitr4i 186	1 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   E.wrex 2418   _Vcvv 2689

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691

This theorem is referenced by: (None)