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Theorem sbciedf 2820
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 112 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 1431 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 sbciegft 2815 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒))) → ([𝐴 / 𝑥]𝜓𝜒))
81, 2, 6, 7syl3anc 1146 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wnf 1365  wcel 1409  [wsbc 2786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787
This theorem is referenced by:  sbcied  2821  sbc2iegf  2855  csbiebt  2913  sbcnestgf  2924  ovmpt2dxf  5653
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