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Theorem csbiegf 2971
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 1383 . 2 𝑥(𝑥 = 𝐴𝐵 = 𝐶)
3 csbiegf.1 . . 3 (𝐴𝑉𝑥𝐶)
4 csbiebt 2967 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
53, 4mpdan 412 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
62, 5mpbii 146 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wcel 1438  wnfc 2215  csb 2933
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  csbief  2972  sbcco3g  2985  csbco3g  2986  fmptcof  5465  fmpt2co  5981  iseqf1olemjpcl  9920  iseqf1olemqpcl  9921  iseqf1olemfvp  9922  seq3f1olemqsum  9925  sumsnf  10799
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