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Theorem csbfv12g 5336
 Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbfv12g (A C[A / x](FB) = ([A / x]F[A / x]B))

Proof of Theorem csbfv12g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbiotag 4371 . . 3 (A C[A / x](℩yBFy) = (℩yA / xBFy))
2 sbcbrg 4685 . . . . 5 (A C → ([̣A / xBFy[A / x]B[A / x]F[A / x]y))
3 csbconstg 3150 . . . . . 6 (A C[A / x]y = y)
43breq2d 4651 . . . . 5 (A C → ([A / x]B[A / x]F[A / x]y[A / x]B[A / x]Fy))
52, 4bitrd 244 . . . 4 (A C → ([̣A / xBFy[A / x]B[A / x]Fy))
65iotabidv 4360 . . 3 (A C → (℩yA / xBFy) = (℩y[A / x]B[A / x]Fy))
71, 6eqtrd 2385 . 2 (A C[A / x](℩yBFy) = (℩y[A / x]B[A / x]Fy))
8 df-fv 4795 . . 3 (FB) = (℩yBFy)
98csbeq2i 3162 . 2 [A / x](FB) = [A / x](℩yBFy)
10 df-fv 4795 . 2 ([A / x]F[A / x]B) = (℩y[A / x]B[A / x]Fy)
117, 9, 103eqtr4g 2410 1 (A C[A / x](FB) = ([A / x]F[A / x]B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  [̣wsbc 3046  [csb 3136  ℩cio 4337   class class class wbr 4639   ‘cfv 4781 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-csb 3137  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 This theorem is referenced by:  csbfv2g  5337
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