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Theorem csbiegf 3890
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 groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbiegf
StepHypRef Expression
1 csbiegf.2 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
21ax-gen 1797 . 2 𝑥(𝑥 = 𝐴𝐵 = 𝐶)
3 csbiegf.1 . . 3 (𝐴𝑉𝑥𝐶)
4 csbiebt 3886 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
53, 4mpdan 686 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
62, 5mpbii 236 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2115  wnfc 2958  csb 3857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-sbc 3750  df-csb 3858
This theorem is referenced by:  csbief  3891  sbcco3gw  4347  sbcco3g  4352  csbco3g  4353  fmptcof  6865  fmpoco  7765  sumsnf  15078  prodsn  15295  prodsnf  15297  bpolylem  15381  pcmpt  16205  chfacfpmmulfsupp  21446  elmptrab  22410  dvfsumrlim3  24614  itgsubstlem  24629  itgsubst  24630  ifeqeqx  30283  disjunsn  30330  sbcaltop  33449  unirep  35029  cdleme31so  37553  cdleme31sn  37554  cdleme31sn1  37555  cdleme31se  37556  cdleme31se2  37557  cdleme31sc  37558  cdleme31sde  37559  cdleme31sn2  37563  cdlemeg47rv2  37684  cdlemk41  38094  monotuz  39677  oddcomabszz  39680
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