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Theorem elimdelov 5573
 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 (φC (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
StepHypRef Expression
1 iftrue 3668 . . . 4 (φ → if(φ, C, Z) = C)
2 elimdelov.1 . . . 4 (φC (AFB))
31, 2eqeltrd 2427 . . 3 (φ → if(φ, C, Z) (AFB))
4 iftrue 3668 . . . 4 (φ → if(φ, A, X) = A)
5 iftrue 3668 . . . 4 (φ → if(φ, B, Y) = B)
64, 5oveq12d 5540 . . 3 (φ → ( if(φ, A, X)F if(φ, B, Y)) = (AFB))
73, 6eleqtrrd 2430 . 2 (φ → if(φ, C, Z) ( if(φ, A, X)F if(φ, B, Y)))
8 iffalse 3669 . . . 4 φ → if(φ, C, Z) = Z)
9 elimdelov.2 . . . 4 Z (XFY)
108, 9syl6eqel 2441 . . 3 φ → if(φ, C, Z) (XFY))
11 iffalse 3669 . . . 4 φ → if(φ, A, X) = X)
12 iffalse 3669 . . . 4 φ → if(φ, B, Y) = Y)
1311, 12oveq12d 5540 . . 3 φ → ( if(φ, A, X)F if(φ, B, Y)) = (XFY))
1410, 13eleqtrrd 2430 . 2 φ → if(φ, C, Z) ( if(φ, A, X)F if(φ, B, Y)))
157, 14pm2.61i 156 1 if(φ, C, Z) ( if(φ, A, X)F if(φ, B, Y))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1710   ifcif 3662  (class class class)co 5525 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526 This theorem is referenced by: (None)
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