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Theorem csbied 3046
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 1508 . 2 𝑥𝜑
2 nfcvd 2282 . 2 (𝜑𝑥𝐶)
3 csbied.1 . 2 (𝜑𝐴𝑉)
4 csbied.2 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
51, 2, 3, 4csbiedf 3040 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  csb 3003
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004
This theorem is referenced by:  csbied2  3047  fvmptd  5502  seq3f1olemp  10282  fsumgcl  11162  fsum3  11163  fsumshftm  11221  fisum0diag2  11223
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