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Theorem csbief 3523
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 3522 . 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 1474  wcel 1976  wnfc 2737  Vcvv 3172  csb 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-csb 3499
This theorem is referenced by:  csbie  3524  csbun  3960  csbin  3961  csbif  4087  csbopab  4922  csbopabgALT  4923  csbima12  5389  csbiota  5783  csbriota  6501  csbov123  6563  pcmpt  15380  mpfrcl  19285  iundisj2  23041  iundisj2f  28591  iundisj2fi  28749  csbdif  32143  sbccom2f  32897  csbcog  36756  csbingOLD  37872  csbafv12g  39664  csbaovg  39707
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