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Theorem sbccsb2 4383
Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbccsb2 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})

Proof of Theorem sbccsb2
StepHypRef Expression
1 sbcex 3779 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 elex 3510 . 2 (𝐴𝐴 / 𝑥{𝑥𝜑} → 𝐴 ∈ V)
3 abid 2800 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43sbcbii 3826 . . 3 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
5 sbcel12 4357 . . . 4 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑})
6 csbvarg 4380 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
76eleq1d 2894 . . . 4 (𝐴 ∈ V → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
85, 7syl5bb 284 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
94, 8syl5bbr 286 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
101, 2, 9pm5.21nii 380 1 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 207  wcel 2105  {cab 2796  Vcvv 3492  [wsbc 3769  csb 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-nul 4289
This theorem is referenced by: (None)
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