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Theorem sbciedf 2858
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 113 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 1456 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 sbciegft 2853 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒))) → ([𝐴 / 𝑥]𝜓𝜒))
81, 2, 6, 7syl3anc 1170 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wnf 1390  wcel 1434  [wsbc 2824
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  sbcied  2859  sbc2iegf  2893  csbiebt  2951  sbcnestgf  2962  ovmpt2dxf  5677
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