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Theorem csbfv12g 5152
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbfv12g (A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))

Proof of Theorem csbfv12g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbiotag 4838 . . 3 (A 𝐶A / x(℩yB𝐹y) = (℩y[A / x]B𝐹y))
2 sbcbrg 3804 . . . . 5 (A 𝐶 → ([A / x]B𝐹yA / xBA / x𝐹A / xy))
3 csbconstg 2858 . . . . . 6 (A 𝐶A / xy = y)
43breq2d 3767 . . . . 5 (A 𝐶 → (A / xBA / x𝐹A / xyA / xBA / x𝐹y))
52, 4bitrd 177 . . . 4 (A 𝐶 → ([A / x]B𝐹yA / xBA / x𝐹y))
65iotabidv 4831 . . 3 (A 𝐶 → (℩y[A / x]B𝐹y) = (℩yA / xBA / x𝐹y))
71, 6eqtrd 2069 . 2 (A 𝐶A / x(℩yB𝐹y) = (℩yA / xBA / x𝐹y))
8 df-fv 4853 . . 3 (𝐹B) = (℩yB𝐹y)
98csbeq2i 2870 . 2 A / x(𝐹B) = A / x(℩yB𝐹y)
10 df-fv 4853 . 2 (A / x𝐹A / xB) = (℩yA / xBA / x𝐹y)
117, 9, 103eqtr4g 2094 1 (A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  [wsbc 2758  csb 2846   class class class wbr 3755  cio 4808  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853
This theorem is referenced by:  csbfv2g  5153
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