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Theorem sbcie2g 3598
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3599 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 3566 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 sbcie2g.2 . 2 (𝑦 = 𝐴 → (𝜓𝜒))
3 sbsbc 3568 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
4 nfv 1980 . . . 4 𝑥𝜓
5 sbcie2g.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5sbie 2533 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
73, 6bitr3i 266 . 2 ([𝑦 / 𝑥]𝜑𝜓)
81, 2, 7vtoclbg 3395 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  [wsb 2034  wcel 2127  [wsbc 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-v 3330  df-sbc 3565
This theorem is referenced by:  sbcel2gv  3625  csbie2g  3693  brab1  4840  bnj90  31068  bnj124  31219  riotasvd  34714
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