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Theorem sbcie2g 3762
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3763 avoids a disjointness condition on 𝑥, 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
sbcie2g.1 (𝑥 = 𝑦 → (𝜑𝜓))
sbcie2g.2 (𝑦 = 𝐴 → (𝜓𝜒))
Assertion
Ref Expression
sbcie2g (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜒,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem sbcie2g
StepHypRef Expression
1 dfsbcq 3725 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 sbcie2g.2 . 2 (𝑦 = 𝐴 → (𝜓𝜒))
3 sbsbc 3727 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
4 sbcie2g.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
54sbievw 2101 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
63, 5bitr3i 280 . 2 ([𝑦 / 𝑥]𝜑𝜓)
71, 2, 6vtoclbg 3520 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  [wsb 2069  wcel 2112  [wsbc 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-sbc 3724
This theorem is referenced by:  sbcel2gv  3790  csbie2g  3871  brab1  5081  bnj90  32106  bnj124  32257  riotasvd  36251
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