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Theorem sbcied2 2823
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied2.1 (𝜑𝐴𝑉)
sbcied2.2 (𝜑𝐴 = 𝐵)
sbcied2.3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
sbcied2 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied2
StepHypRef Expression
1 sbcied2.1 . 2 (𝜑𝐴𝑉)
2 id 19 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 sbcied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2110 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 sbcied2.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
64, 5syldan 270 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
71, 6sbcied 2822 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  [wsbc 2787
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 2788
This theorem is referenced by: (None)
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