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Theorem sbcalfi 35275
Description: Move universal quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
sbcalfi.1 𝑦𝐴
sbcalfi.2 ([𝐴 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbcalfi ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem sbcalfi
StepHypRef Expression
1 sbcalfi.1 . . 3 𝑦𝐴
21sbcalf 35273 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
3 sbcalfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43albii 1811 . 2 (∀𝑦[𝐴 / 𝑥]𝜑 ↔ ∀𝑦𝜓)
52, 4bitri 276 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1526  wnfc 2958  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sbc 3770
This theorem is referenced by: (None)
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