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Theorem sbcie2g 3079
 Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3080 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 (A V → ([̣A / xφχ))
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 3048 . 2 (y = A → ([̣y / xφ ↔ [̣A / xφ))
2 sbcie2g.2 . 2 (y = A → (ψχ))
3 sbsbc 3050 . . 3 ([y / x]φ ↔ [̣y / xφ)
4 nfv 1619 . . . 4 xψ
5 sbcie2g.1 . . . 4 (x = y → (φψ))
64, 5sbie 2038 . . 3 ([y / x]φψ)
73, 6bitr3i 242 . 2 ([̣y / xφψ)
81, 2, 7vtoclbg 2915 1 (A V → ([̣A / xφχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  csbie2g  3182  brab1  4684
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