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Theorem sbcalf 38101
Description: Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Hypothesis
Ref Expression
sbcalf.1 𝑦𝐴
Assertion
Ref Expression
sbcalf ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem sbcalf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sb8v 2353 . . 3 (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
21sbcbii 3852 . 2 ([𝐴 / 𝑥]𝑦𝜑[𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑)
3 sbcal 3855 . 2 ([𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
4 sbcalf.1 . . . 4 𝑦𝐴
5 nfs1v 2154 . . . 4 𝑦[𝑧 / 𝑦]𝜑
64, 5nfsbcw 3813 . . 3 𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
7 nfv 1912 . . 3 𝑧[𝐴 / 𝑥]𝜑
8 sbequ12r 2250 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
98sbcbidv 3851 . . 3 (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥]𝜑))
106, 7, 9cbvalv1 2342 . 2 (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
112, 3, 103bitri 297 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1535  [wsb 2062  wnfc 2888  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792
This theorem is referenced by:  sbcalfi  38103
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