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Theorem sbcie2g 3814
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3815 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 3774 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 sbcie2g.2 . 2 (𝑦 = 𝐴 → (𝜓𝜒))
3 sbsbc 3776 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
4 sbcie2g.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
54sbievw 2087 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
63, 5bitr3i 277 . 2 ([𝑦 / 𝑥]𝜑𝜓)
71, 2, 6vtoclbg 3539 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  [wsb 2059  wcel 2098  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773
This theorem is referenced by:  sbcel2gv  3844  csbie2g  3931  brab1  5189  bnj90  34261  bnj124  34410  riotasvd  38338
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