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Theorem sbcal 2960
 Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
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 2917 . 2 ([𝐴 / 𝑦]𝑥𝜑𝐴 ∈ V)
2 sbcex 2917 . . 3 ([𝐴 / 𝑦]𝜑𝐴 ∈ V)
32sps 1517 . 2 (∀𝑥[𝐴 / 𝑦]𝜑𝐴 ∈ V)
4 dfsbcq2 2912 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑[𝐴 / 𝑦]𝑥𝜑))
5 dfsbcq2 2912 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑[𝐴 / 𝑦]𝜑))
65albidv 1796 . . 3 (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7 sbal 1975 . . 3 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
84, 6, 7vtoclbg 2747 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
91, 3, 8pm5.21nii 693 1 ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∀wal 1329   = wceq 1331   ∈ wcel 1480  [wsb 1735  Vcvv 2686  [wsbc 2909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910 This theorem is referenced by:  sbcfung  5147
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