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Theorem sbcalf 35428
Description: Move universal quantifier in and out of class substitution, with an explicit non-free 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 nfv 1915 . . . 4 𝑧𝜑
21sb8v 2373 . . 3 (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
32sbcbii 3808 . 2 ([𝐴 / 𝑥]𝑦𝜑[𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑)
4 sbcal 3812 . 2 ([𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 sbcalf.1 . . . 4 𝑦𝐴
6 nfs1v 2160 . . . 4 𝑦[𝑧 / 𝑦]𝜑
75, 6nfsbcw 3774 . . 3 𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8 nfv 1915 . . 3 𝑧[𝐴 / 𝑥]𝜑
9 sbequ12r 2254 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
109sbcbidv 3806 . . 3 (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥]𝜑))
117, 8, 10cbvalv1 2361 . 2 (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
123, 4, 113bitri 299 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wal 1535  [wsb 2069  wnfc 2957  [wsbc 3752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-sbc 3753
This theorem is referenced by:  sbcalfi  35430
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