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Theorem csbfv12 6198
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 5850 . . . 4 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦)
2 sbcbr123 4676 . . . . . 6 ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦)
3 csbconstg 3532 . . . . . . 7 (𝐴 ∈ V → 𝐴 / 𝑥𝑦 = 𝑦)
43breq2d 4635 . . . . . 6 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
52, 4syl5bb 272 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
65iotabidv 5841 . . . 4 (𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
71, 6syl5eq 2667 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
8 df-fv 5865 . . . 4 (𝐹𝐵) = (℩𝑦𝐵𝐹𝑦)
98csbeq2i 3971 . . 3 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦)
10 df-fv 5865 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦)
117, 9, 103eqtr4g 2680 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
12 csbprc 3958 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = ∅)
13 csbprc 3958 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐹 = ∅)
1413fveq1d 6160 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (∅‘𝐴 / 𝑥𝐵))
15 0fv 6194 . . . 4 (∅‘𝐴 / 𝑥𝐵) = ∅
1614, 15syl6req 2672 . . 3 𝐴 ∈ V → ∅ = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
1712, 16eqtrd 2655 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
1811, 17pm2.61i 176 1 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3190  [wsbc 3422  csb 3519  c0 3897   class class class wbr 4623  cio 5818  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759  ax-pow 4813
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-dm 5094  df-iota 5820  df-fv 5865
This theorem is referenced by:  csbfv2g  6199  coe1fzgsumdlem  19611  evl1gsumdlem  19660  csbwrecsg  32844  csbrdgg  32846  rdgeqoa  32889  csbfinxpg  32896  cdlemk42  35748  iccelpart  40697
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