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Theorem csbfv12g 5303
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbfv12g (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Proof of Theorem csbfv12g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbiotag 4974 . . 3 (𝐴𝐶𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦))
2 sbcbrg 3869 . . . . 5 (𝐴𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
3 csbconstg 2934 . . . . . 6 (𝐴𝐶𝐴 / 𝑥𝑦 = 𝑦)
43breq2d 3832 . . . . 5 (𝐴𝐶 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
52, 4bitrd 186 . . . 4 (𝐴𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
65iotabidv 4967 . . 3 (𝐴𝐶 → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
71, 6eqtrd 2117 . 2 (𝐴𝐶𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
8 df-fv 4989 . . 3 (𝐹𝐵) = (℩𝑦𝐵𝐹𝑦)
98csbeq2i 2946 . 2 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦)
10 df-fv 4989 . 2 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦)
117, 9, 103eqtr4g 2142 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  [wsbc 2829  csb 2922   class class class wbr 3820  cio 4944  cfv 4981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-iota 4946  df-fv 4989
This theorem is referenced by:  csbfv2g  5304
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