Theorem elimdelov 5574

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

Hypotheses

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

Assertion

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

Proof of Theorem elimdelov

Step	Hyp	Ref	Expression
1		iftrue 3669	. . . 4 ⊢ (φ → if(φ, C, Z) = C)
2		elimdelov.1	. . . 4 ⊢ (φ → C ∈ (AFB))
3	1, 2	eqeltrd 2427	. . 3 ⊢ (φ → if(φ, C, Z) ∈ (AFB))
4		iftrue 3669	. . . 4 ⊢ (φ → if(φ, A, X) = A)
5		iftrue 3669	. . . 4 ⊢ (φ → if(φ, B, Y) = B)
6	4, 5	oveq12d 5541	. . 3 ⊢ (φ → ( if(φ, A, X)F if(φ, B, Y)) = (AFB))
7	3, 6	eleqtrrd 2430	. 2 ⊢ (φ → if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y)))
8		iffalse 3670	. . . 4 ⊢ (¬ φ → if(φ, C, Z) = Z)
9		elimdelov.2	. . . 4 ⊢ Z ∈ (XFY)
10	8, 9	syl6eqel 2441	. . 3 ⊢ (¬ φ → if(φ, C, Z) ∈ (XFY))
11		iffalse 3670	. . . 4 ⊢ (¬ φ → if(φ, A, X) = X)
12		iffalse 3670	. . . 4 ⊢ (¬ φ → if(φ, B, Y) = Y)
13	11, 12	oveq12d 5541	. . 3 ⊢ (¬ φ → ( if(φ, A, X)F if(φ, B, Y)) = (XFY))
14	10, 13	eleqtrrd 2430	. 2 ⊢ (¬ φ → if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y)))
15	7, 14	pm2.61i 156	1 ⊢ if(φ, C, Z) ∈ ( if(φ, A, X)F if(φ, B, Y))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1710   ifcif 3663  (class class class)co 5526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796  df-ov 5527

This theorem is referenced by: (None)