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Theorem csbief 3929
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbief.1 𝐴 ∈ V
csbief.2 𝑥𝐶
csbief.3 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbief 𝐴 / 𝑥𝐵 = 𝐶
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbief
StepHypRef Expression
1 csbief.1 . 2 𝐴 ∈ V
2 csbief.2 . . . 4 𝑥𝐶
32a1i 11 . . 3 (𝐴 ∈ V → 𝑥𝐶)
4 csbief.3 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3928 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
61, 5ax-mp 5 1 𝐴 / 𝑥𝐵 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wnfc 2884  Vcvv 3475  csb 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-sbc 3779  df-csb 3895
This theorem is referenced by:  csbieOLD  3931  cbvrabcsfw  3938  csbun  4439  csbin  4440  csbdif  4528  csbif  4586  csbopab  5556  csbopabgALT  5557  csbima12  6079  csbcog  6297  csbiota  6537  csbriota  7381  csbov123  7451  pcmpt  16825  mpfrcl  21648  iundisj2  25066  iundisj2f  31821  iundisj2fi  32008  csbafv12g  45845  csbaovg  45888  csbafv212g  45927
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