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Theorem sbccsb2g 3002
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 2105 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21sbcbii 2940 . 2 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel12g 2988 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑}))
4 csbvarg 3000 . . . 4 (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
54eleq1d 2186 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
63, 5bitrd 187 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
72, 6syl5bbr 193 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1465  {cab 2103  [wsbc 2882  csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by: (None)
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