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Theorem sbcie2g 3717
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3718 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 3685 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 sbcie2g.2 . 2 (𝑦 = 𝐴 → (𝜓𝜒))
3 sbsbc 3687 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
4 sbcie2g.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
54sbievw 2041 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
63, 5bitr3i 269 . 2 ([𝑦 / 𝑥]𝜑𝜓)
71, 2, 6vtoclbg 3487 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  [wsb 2015  wcel 2050  [wsbc 3683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-sbc 3684
This theorem is referenced by:  sbcel2gv  3746  csbie2g  3821  brab1  4978  bnj90  31640  bnj124  31790  riotasvd  35537
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