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Theorem sbciedf 3816
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 415 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 2212 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 sbciegft 3811 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒))) → ([𝐴 / 𝑥]𝜓𝜒))
81, 2, 6, 7syl3anc 1367 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1534   = wceq 1536  wnf 1783  wcel 2113  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3499  df-sbc 3776
This theorem is referenced by:  sbcied  3817  sbc2iegf  3852  csbiebt  3915  sbcnestgfw  4373  sbcnestgf  4378  ovmpodxf  7303  sbc2iedf  30233  reuf1odnf  43313  ovmpordxf  44394
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