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Theorem sbccsb2g 3027
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 2125 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21sbcbii 2963 . 2 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel12g 3012 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑}))
4 csbvarg 3025 . . . 4 (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
54eleq1d 2206 . . 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 1480  {cab 2123  [wsbc 2904  csb 2998
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999
This theorem is referenced by: (None)
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