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Theorem sbciedf 3438
 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 449 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 2069 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 sbciegft 3433 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒))) → ([𝐴 / 𝑥]𝜓𝜒))
81, 2, 6, 7syl3anc 1318 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  sbcied  3439  sbc2iegf  3471  csbiebt  3519  sbcnestgf  3947  ovmpt2dxf  6684  ovmpt2rdxf  41910
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