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Theorem sbccsb2 4201
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 3641 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 elex 3404 . 2 (𝐴𝐴 / 𝑥{𝑥𝜑} → 𝐴 ∈ V)
3 abid 2792 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43sbcbii 3687 . . 3 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
5 sbcel12 4178 . . . 4 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑})
6 csbvarg 4198 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
76eleq1d 2868 . . . 4 (𝐴 ∈ V → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
85, 7syl5bb 274 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
94, 8syl5bbr 276 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
101, 2, 9pm5.21nii 369 1 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2156  {cab 2790  Vcvv 3389  [wsbc 3631  csb 3726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-nul 4115
This theorem is referenced by: (None)
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