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Theorem sbciedf 2944
 Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1 (𝜑𝐴𝑉)
sbcied.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
sbciedf.3 𝑥𝜑
sbciedf.4 (𝜑 → Ⅎ𝑥𝜒)
Assertion
Ref Expression
sbciedf (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝑉(𝑥)

Proof of Theorem sbciedf
StepHypRef Expression
1 sbcied.1 . 2 (𝜑𝐴𝑉)
2 sbciedf.4 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 sbciedf.3 . . 3 𝑥𝜑
4 sbcied.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
54ex 114 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 1502 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 sbciegft 2939 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒))) → ([𝐴 / 𝑥]𝜓𝜒))
81, 2, 6, 7syl3anc 1216 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  [wsbc 2909 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 This theorem is referenced by:  sbcied  2945  sbc2iegf  2979  csbiebt  3039  sbcnestgf  3051  ovmpodxf  5896
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