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Theorem csbfv12g 5389
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 5052 . . 3 (𝐴𝐶𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦))
2 sbcbrg 3924 . . . . 5 (𝐴𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
3 csbconstg 2967 . . . . . 6 (𝐴𝐶𝐴 / 𝑥𝑦 = 𝑦)
43breq2d 3887 . . . . 5 (𝐴𝐶 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
52, 4bitrd 187 . . . 4 (𝐴𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
65iotabidv 5045 . . 3 (𝐴𝐶 → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
71, 6eqtrd 2132 . 2 (𝐴𝐶𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦))
8 df-fv 5067 . . 3 (𝐹𝐵) = (℩𝑦𝐵𝐹𝑦)
98csbeq2i 2979 . 2 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(℩𝑦𝐵𝐹𝑦)
10 df-fv 5067 . 2 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = (℩𝑦𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝑦)
117, 9, 103eqtr4g 2157 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  [wsbc 2862  csb 2955   class class class wbr 3875  cio 5022  cfv 5059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067
This theorem is referenced by:  csbfv2g  5390
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