Theorem rexcom4b 2785

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 2784	. 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 2768	. . . 4 |- E. x x = B
4	3	biantru 302	. . 3 |- ( ph <-> ( ph /\ E. x x = B ) )
5	4	rexbii 2501	. 2 |- ( E. y e. A ph <-> E. y e. A ( ph /\ E. x x = B ) )
6	1, 5	bitr4i 187	1 |- ( E. x E. y e. A ( ph /\ x = B ) <-> E. y e. A ph )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   _Vcvv 2760

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762

This theorem is referenced by: (None)