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Theorem csbiegf 3043
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 1425 . 2 𝑥(𝑥 = 𝐴𝐵 = 𝐶)
3 csbiegf.1 . . 3 (𝐴𝑉𝑥𝐶)
4 csbiebt 3039 . . 3 ((𝐴𝑉𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
53, 4mpdan 417 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
62, 5mpbii 147 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329   = wceq 1331  wcel 1480  wnfc 2268  csb 3003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004
This theorem is referenced by:  csbief  3044  sbcco3g  3057  csbco3g  3058  fmptcof  5587  fmpoco  6113  iseqf1olemjpcl  10275  iseqf1olemqpcl  10276  iseqf1olemfvp  10277  seq3f1olemqsum  10280  sumsnf  11185
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