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

Theorem sbcal 3837
Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcal ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem sbcal
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3783 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3783 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32sps 2173 . 2 (∀𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3776 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3776 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65albidv 1915 . . 3 (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7 sbal 2158 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3535 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 377 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531   = wceq 1533  [wsb 2059  wcel 2098  Vcvv 3461  [wsbc 3773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-sbc 3774
This theorem is referenced by:  sbcabel  3868  sbcssg  4525  sbcfung  6578  bnj89  34483  bnj110  34620  bnj611  34680  bnj1000  34703  bj-sbeq  36510  bj-sbceqgALT  36511  sbcalf  37718  frege70  43505  frege77  43512  frege116  43551  frege118  43553  trsbc  44121  trsbcVD  44458  sbcssgVD  44464
  Copyright terms: Public domain W3C validator