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Theorem sbcie2g 2848
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2849 avoids a disjointness condition on 𝑥 and 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 2818 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 sbcie2g.2 . 2 (𝑦 = 𝐴 → (𝜓𝜒))
3 sbsbc 2820 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
4 nfv 1462 . . . 4 𝑥𝜓
5 sbcie2g.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5sbie 1715 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
73, 6bitr3i 184 . 2 ([𝑦 / 𝑥]𝜑𝜓)
81, 2, 7vtoclbg 2660 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  [wsb 1686  [wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by:  sbcel2gv  2878  csbie2g  2953  brab1  3838
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