Theorem elimdeloprv 3940

Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 8656 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Hypotheses

Ref	Expression
elimdeloprv.1	|- (ph -> C e. (AFB))
elimdeloprv.2	|- Z e. (XFY)

Assertion

Ref	Expression
elimdeloprv	|- if(ph, C, Z) e. (if(ph, A, X)Fif(ph, B, Y))

Proof of Theorem elimdeloprv

Step	Hyp	Ref	Expression
1		iftrue 2337	. . . 4 |- (ph -> if(ph, C, Z) = C)
2		elimdeloprv.1	. . . 4 |- (ph -> C e. (AFB))
3	1, 2	eqeltrd 1524	. . 3 |- (ph -> if(ph, C, Z) e. (AFB))
4		iftrue 2337	. . . 4 |- (ph -> if(ph, A, X) = A)
5		iftrue 2337	. . . 4 |- (ph -> if(ph, B, Y) = B)
6	4, 5	opreq12d 3917	. . 3 |- (ph -> (if(ph, A, X)Fif(ph, B, Y)) = (AFB))
7	3, 6	eleqtrrd 1527	. 2 |- (ph -> if(ph, C, Z) e. (if(ph, A, X)Fif(ph, B, Y)))
8		iffalse 2338	. . . 4 |- (-. ph -> if(ph, C, Z) = Z)
9		elimdeloprv.2	. . . 4 |- Z e. (XFY)
10	8, 9	syl6eqel 1532	. . 3 |- (-. ph -> if(ph, C, Z) e. (XFY))
11		iffalse 2338	. . . 4 |- (-. ph -> if(ph, A, X) = X)
12		iffalse 2338	. . . 4 |- (-. ph -> if(ph, B, Y) = Y)
13	11, 12	opreq12d 3917	. . 3 |- (-. ph -> (if(ph, A, X)Fif(ph, B, Y)) = (XFY))
14	10, 13	eleqtrrd 1527	. 2 |- (-. ph -> if(ph, C, Z) e. (if(ph, A, X)Fif(ph, B, Y)))
15	7, 14	pm2.61i 126	1 |- if(ph, C, Z) e. (if(ph, A, X)Fif(ph, B, Y))

Syntax hints:  -. wn 2   -> wi 3   e. wcel 1105  ifcif 2332  (class class class)co 3902

This theorem is referenced by:  ghomgrplem 8656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-xp 3147  df-cnv 3149  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fv 3161  df-opr 3904

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