Theorem rexcom4b 10148

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.xE.y e. A (ph /\ x = B) <-> E.y e. A ph)

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

Proof of Theorem rexcom4b

Step	Hyp	Ref	Expression
1		rexcom4a 10147	. 2 |- (E.xE.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 2297	. . . 4 |- E.x x = B
4	3	biantru 793	. . 3 |- (ph <-> (ph /\ E.x x = B))
5	4	rexbii 2128	. 2 |- (E.y e. A ph <-> E.y e. A (ph /\ E.x x = B))
6	1, 5	bitr4i 193	1 |- (E.xE.y e. A (ph /\ x = B) <-> E.y e. A ph)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294

Copyright terms: Public domain