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Theorem sbcal 3831
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 3780 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3780 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32sps 2176 . 2 (∀𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3773 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3773 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65albidv 1914 . . 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 3567 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 382 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wal 1528   = wceq 1530  [wsb 2062  wcel 2107  Vcvv 3493  [wsbc 3770
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-v 3495  df-sbc 3771
This theorem is referenced by:  sbcabel  3859  sbcssg  4461  sbcfung  6372  bnj89  31979  bnj110  32118  bnj611  32178  bnj1000  32201  bj-sbeq  34206  bj-sbceqgALT  34207  sbcalf  35379  frege70  40264  frege77  40271  frege116  40310  frege118  40312  trsbc  40859  trsbcVD  41196  sbcssgVD  41202
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