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Theorem sbcalfi 37495
Description: Move universal quantifier in and out of class substitution, with an explicit nonfree 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 37493 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
3 sbcalfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43albii 1813 . 2 (∀𝑦[𝐴 / 𝑥]𝜑 ↔ ∀𝑦𝜓)
52, 4bitri 275 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531  wnfc 2877  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-sbc 3773
This theorem is referenced by: (None)
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