Theorem ifexg 13599

Description: Conditional operator existence. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
ifexg	|- ((A e. C /\ B e. D) -> if(ph, A, B) e. _V)

Proof of Theorem ifexg

Step	Hyp	Ref	Expression
1		visset 2295	. . . . 5 |- x e. _V
2		visset 2295	. . . . 5 |- y e. _V
3	1, 2	ifex 3031	. . . 4 |- if(ph, x, y) e. _V
4	3	a1i 8	. . 3 |- ((x e. C /\ y e. D) -> if(ph, x, y) e. _V)
5	4	rgen2 2186	. 2 |- A.x e. C A.y e. D if(ph, x, y) e. _V
6		ifeq1 2985	. . . 4 |- (x = A -> if(ph, x, y) = if(ph, A, y))
7	6	eleq1d 1963	. . 3 |- (x = A -> (if(ph, x, y) e. _V <-> if(ph, A, y) e. _V))
8		ifeq2 2987	. . . 4 |- (y = B -> if(ph, A, y) = if(ph, A, B))
9	8	eleq1d 1963	. . 3 |- (y = B -> (if(ph, A, y) e. _V <-> if(ph, A, B) e. _V))
10	7, 9	rcla42v 2384	. 2 |- ((A e. C /\ B e. D) -> (A.x e. C A.y e. D if(ph, x, y) e. _V -> if(ph, A, B) e. _V))
11	5, 10	mpi 55	1 |- ((A e. C /\ B e. D) -> if(ph, A, B) e. _V)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  ifcif 2982

This theorem is referenced by:  eucalgval 13749

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-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-if 2983

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