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Theorem sbcal 3849
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 3798 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 3798 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32sps 2185 . 2 (∀𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 3791 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 3791 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65albidv 1920 . . 3 (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7 sbal 2169 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 3557 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 378 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  [wsb 2064  wcel 2108  Vcvv 3480  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sbc 3789
This theorem is referenced by:  sbcabel  3878  sbcssg  4520  sbcfung  6590  bnj89  34735  bnj110  34872  bnj611  34932  bnj1000  34955  bj-sbeq  36902  bj-sbceqgALT  36903  sbcalf  38121  frege70  43946  frege77  43953  frege116  43992  frege118  43994  trsbc  44560  trsbcVD  44897  sbcssgVD  44903
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