Theorem rexcom4b 2788

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 2787	. 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 2771	. . . 4 |- E. x x = B
4	3	biantru 302	. . 3 |- ( ph <-> ( ph /\ E. x x = B ) )
5	4	rexbii 2504	. 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 1506   e. wcel 2167   E.wrex 2476   _Vcvv 2763

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765

This theorem is referenced by: (None)