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Theorem csbiegf 3941
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiegf.1 (𝐴𝑉𝑥𝐶)
csbiegf.2 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbiegf (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiegf
StepHypRef Expression
1 csbiegf.2 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
21ax-gen 1791 . 2 𝑥(𝑥 = 𝐴𝐵 = 𝐶)
3 csbiegf.1 . . 3 (𝐴𝑉𝑥𝐶)
4 csbiebt 3937 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
53, 4mpdan 687 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
62, 5mpbii 233 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534   = wceq 1536  wcel 2105  wnfc 2887  csb 3907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-v 3479  df-sbc 3791  df-csb 3908
This theorem is referenced by:  csbief  3942  sbcco3gw  4430  sbcco3g  4435  csbco3g  4436  fmptcof  7149  fmpoco  8118  sumsnf  15775  prodsn  15994  prodsnf  15996  bpolylem  16080  pcmpt  16925  chfacfpmmulfsupp  22884  elmptrab  23850  dvfsumrlim3  26088  itgsubstlem  26103  itgsubst  26104  ifeqeqx  32562  disjunsn  32613  sbcaltop  35962  unirep  37700  cdleme31so  40361  cdleme31sn  40362  cdleme31sn1  40363  cdleme31se  40364  cdleme31se2  40365  cdleme31sc  40366  cdleme31sde  40367  cdleme31sn2  40371  cdlemeg47rv2  40492  cdlemk41  40902  monotuz  42929  oddcomabszz  42932
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