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Theorem sbccsb2 3979
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 3428 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 elex 3198 . 2 (𝐴𝐴 / 𝑥{𝑥𝜑} → 𝐴 ∈ V)
3 abid 2609 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43sbcbii 3474 . . 3 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
5 sbcel12 3957 . . . 4 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑})
6 csbvarg 3977 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
76eleq1d 2683 . . . 4 (𝐴 ∈ V → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
85, 7syl5bb 272 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
94, 8syl5bbr 274 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
101, 2, 9pm5.21nii 368 1 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  {cab 2607  Vcvv 3186  [wsbc 3418  csb 3515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-nul 3894
This theorem is referenced by: (None)
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