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Theorem sbcalf 34047
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 1883 . . . 4 𝑧𝜑
21sb8 2452 . . 3 (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
32sbcbii 3524 . 2 ([𝐴 / 𝑥]𝑦𝜑[𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑)
4 sbcal 3518 . 2 ([𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 sbcalf.1 . . . 4 𝑦𝐴
6 nfs1v 2465 . . . 4 𝑦[𝑧 / 𝑦]𝜑
75, 6nfsbc 3490 . . 3 𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8 nfv 1883 . . 3 𝑧[𝐴 / 𝑥]𝜑
9 sbequ12r 2150 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
109sbcbidv 3523 . . 3 (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥]𝜑))
117, 8, 10cbval 2307 . 2 (∀𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
123, 4, 113bitri 286 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521  [wsb 1937  wnfc 2780  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469
This theorem is referenced by:  sbcalfi  34049
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