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Theorem csbied 3087
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 1515 . 2 𝑥𝜑
2 nfcvd 2307 . 2 (𝜑𝑥𝐶)
3 csbied.1 . 2 (𝜑𝐴𝑉)
4 csbied.2 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
51, 2, 3, 4csbiedf 3081 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1342  wcel 2135  csb 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-sbc 2948  df-csb 3042
This theorem is referenced by:  csbied2  3088  fvmptd  5562  seq3f1olemp  10428  fsumgcl  11317  fsum3  11318  fsumshftm  11376  fisum0diag2  11378  fprodseq  11514  fprodeq0  11548
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