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Theorem sbcalf 34404
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 2010 . . . 4 𝑧𝜑
21sb8 2542 . . 3 (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
32sbcbii 3689 . 2 ([𝐴 / 𝑥]𝑦𝜑[𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑)
4 sbcal 3683 . 2 ([𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 sbcalf.1 . . . 4 𝑦𝐴
6 nfs1v 2303 . . . 4 𝑦[𝑧 / 𝑦]𝜑
75, 6nfsbc 3655 . . 3 𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8 nfv 2010 . . 3 𝑧[𝐴 / 𝑥]𝜑
9 sbequ12r 2279 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
109sbcbidv 3688 . . 3 (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥]𝜑))
117, 8, 10cbvalv1 2357 . 2 (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
123, 4, 113bitri 289 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1651  [wsb 2064  wnfc 2928  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634
This theorem is referenced by:  sbcalfi  34406
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