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Theorem sbcal 3472
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 3432 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3432 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32sps 2053 . 2 (∀𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3425 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3425 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65albidv 1846 . . 3 (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7 sbal 2461 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3257 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 368 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478   = wceq 1480  [wsb 1877  wcel 1987  Vcvv 3190  [wsbc 3422
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-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3192  df-sbc 3423
This theorem is referenced by:  sbcabel  3503  sbcssg  4063  sbcfung  5881  bnj89  30548  bnj538OLD  30571  bnj110  30689  bnj611  30749  bnj1000  30772  bj-sbeq  32596  bj-sbceqgALT  32597  sbcalf  33588  frege70  37748  frege77  37755  frege116  37794  frege118  37796  trsbc  38271
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