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Theorem sbcie2g 2790
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2791 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 (A 𝑉 → ([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 2760 . 2 (y = A → ([y / x]φ[A / x]φ))
2 sbcie2g.2 . 2 (y = A → (ψχ))
3 sbsbc 2762 . . 3 ([y / x]φ[y / x]φ)
4 nfv 1418 . . . 4 xψ
5 sbcie2g.1 . . . 4 (x = y → (φψ))
64, 5sbie 1671 . . 3 ([y / x]φψ)
73, 6bitr3i 175 . 2 ([y / x]φψ)
81, 2, 7vtoclbg 2608 1 (A 𝑉 → ([A / x]φχ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  [wsb 1642  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  sbcel2gv  2816  csbie2g  2890  brab1  3800
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