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Theorem csbied 2974
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbied.1 (𝜑𝐴𝑉)
csbied.2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
csbied (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbied
StepHypRef Expression
1 nfv 1466 . 2 𝑥𝜑
2 nfcvd 2229 . 2 (𝜑𝑥𝐶)
3 csbied.1 . 2 (𝜑𝐴𝑉)
4 csbied.2 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
51, 2, 3, 4csbiedf 2968 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  csbied2  2975  fvmptd  5385  seq3f1olemp  9927  fsumgcl  10773  fisum  10774  fsumshftm  10835  fisum0diag2  10837
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