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Theorem csbief 3699
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 3698 . 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 1632  wcel 2139  wnfc 2889  Vcvv 3340  csb 3674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675
This theorem is referenced by:  csbie  3700  csbun  4152  csbin  4153  csbif  4282  csbopab  5158  csbopabgALT  5159  csbima12  5641  csbiota  6042  csbriota  6787  csbov123  6851  pcmpt  15818  mpfrcl  19740  iundisj2  23537  iundisj2f  29731  iundisj2fi  29886  csbdif  33500  sbccom2f  34262  csbcog  38461  csbingOLD  39572  csbafv12g  41741  csbaovg  41784
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