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Theorem csbief 3753
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 3752 . 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 1653  wcel 2157  wnfc 2928  Vcvv 3385  csb 3728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634  df-csb 3729
This theorem is referenced by:  csbie  3754  csbun  4205  csbin  4206  csbif  4332  csbopab  5204  csbopabgALT  5205  csbima12  5700  csbiota  6094  csbriota  6851  csbov123  6919  pcmpt  15929  mpfrcl  19840  iundisj2  23657  iundisj2f  29920  iundisj2fi  30074  csbdif  33670  sbccom2f  34417  csbcog  38724  csbafv12g  41991  csbaovg  42034  csbafv212g  42073
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