MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbccsb2 Structured version   Visualization version   GIF version

Theorem sbccsb2 4430
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 3783 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 elex 3491 . 2 (𝐴𝐴 / 𝑥{𝑥𝜑} → 𝐴 ∈ V)
3 abid 2712 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43sbcbii 3833 . . 3 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
5 sbcel12 4404 . . . 4 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑})
6 csbvarg 4427 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
76eleq1d 2817 . . . 4 (𝐴 ∈ V → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
85, 7bitrid 282 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
94, 8bitr3id 284 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
101, 2, 9pm5.21nii 379 1 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2106  {cab 2708  Vcvv 3473  [wsbc 3773  csb 3889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-nul 4319
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator