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Theorem sbccsb2g 2936
Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
Assertion
Ref Expression
sbccsb2g (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))

Proof of Theorem sbccsb2g
StepHypRef Expression
1 abid 2070 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21sbcbii 2874 . 2 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel12g 2922 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑}))
4 csbvarg 2934 . . . 4 (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
54eleq1d 2148 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
63, 5bitrd 186 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
72, 6syl5bbr 192 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1434  {cab 2068  [wsbc 2816  csb 2909
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  df-csb 2910
This theorem is referenced by: (None)
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