Theorem elimdeloprv 3986

Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 10294 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Hypotheses

Ref	Expression
elimdeloprv.1	⊢ (φ → C ∈ (AFB))
elimdeloprv.2	⊢ Z ∈ (XFY)

Assertion

Ref	Expression
elimdeloprv	⊢ if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y))

Proof of Theorem elimdeloprv

Step	Hyp	Ref	Expression
1		iftrue 2356	. . . 4 ⊢ (φ → if(φ, C, Z) = C)
2		elimdeloprv.1	. . . 4 ⊢ (φ → C ∈ (AFB))
3	1, 2	eqeltrd 1540	. . 3 ⊢ (φ → if(φ, C, Z) ∈ (AFB))
4		iftrue 2356	. . . 4 ⊢ (φ → if(φ, A, X) = A)
5		iftrue 2356	. . . 4 ⊢ (φ → if(φ, B, Y) = B)
6	4, 5	opreq12d 3963	. . 3 ⊢ (φ → ( if(φ, A, X)F if(φ, B, Y)) = (AFB))
7	3, 6	eleqtrrd 1543	. 2 ⊢ (φ → if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y)))
8		iffalse 2357	. . . 4 ⊢ (¬ φ → if(φ, C, Z) = Z)
9		elimdeloprv.2	. . . 4 ⊢ Z ∈ (XFY)
10	8, 9	syl6eqel 1548	. . 3 ⊢ (¬ φ → if(φ, C, Z) ∈ (XFY))
11		iffalse 2357	. . . 4 ⊢ (¬ φ → if(φ, A, X) = X)
12		iffalse 2357	. . . 4 ⊢ (¬ φ → if(φ, B, Y) = Y)
13	11, 12	opreq12d 3963	. . 3 ⊢ (¬ φ → ( if(φ, A, X)F if(φ, B, Y)) = (XFY))
14	10, 13	eleqtrrd 1543	. 2 ⊢ (¬ φ → if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y)))
15	7, 14	pm2.61i 126	1 ⊢ if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y))

Syntax hints:  ¬ wn 2   → wi 3   ∈ wcel 955   ifcif 2351  (class class class)co 3948

This theorem is referenced by:  ghomgrplem 10294

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950

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