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Theorem csbfv12 6929
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbfv12 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)

Proof of Theorem csbfv12
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbiota 6528 . . . 4 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦)
2 sbcbr123 5198 . . . . . 6 ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦)
3 csbconstg 3910 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥𝑦 = 𝑦)
43breq2d 5156 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
52, 4bitrid 283 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
65iotabidv 6519 . . . 4 (𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
71, 6eqtrid 2785 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
8 df-fv 6543 . . . 4 (𝐹𝐵) = (℩𝑦𝐵𝐹𝑦)
98csbeq2i 3899 . . 3 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦)
10 df-fv 6543 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦)
117, 9, 103eqtr4g 2798 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
12 csbprc 4404 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
13 csbprc 4404 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
1413fveq1d 6883 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (∅‘𝐴 / 𝑥𝐵))
15 0fv 6925 . . . 4 (∅‘𝐴 / 𝑥𝐵) = ∅
1614, 15eqtr2di 2790 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
1712, 16eqtrd 2773 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
1811, 17pm2.61i 182 1 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  [wsbc 3775  csb 3891  c0 4320   class class class wbr 5144  cio 6485  cfv 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-dm 5682  df-iota 6487  df-fv 6543
This theorem is referenced by:  csbfv2g  6930  coe1fzgsumdlem  21794  evl1gsumdlem  21844  csbrdgg  36115  rdgeqoa  36156  csbfinxpg  36174  cdlemk42  39718  iccelpart  45974
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