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Theorem alrimii 35278
Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
alrimii.1 𝑦𝜑
alrimii.2 (𝜑𝜓)
alrimii.3 ([𝑦 / 𝑥]𝜒𝜓)
alrimii.4 𝑦𝜒
Assertion
Ref Expression
alrimii (𝜑 → ∀𝑥𝜒)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem alrimii
StepHypRef Expression
1 alrimii.1 . . 3 𝑦𝜑
2 alrimii.2 . . . 4 (𝜑𝜓)
3 alrimii.3 . . . 4 ([𝑦 / 𝑥]𝜒𝜓)
42, 3sylibr 235 . . 3 (𝜑[𝑦 / 𝑥]𝜒)
51, 4alrimi 2203 . 2 (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜒)
6 nfsbc1v 3789 . . 3 𝑥[𝑦 / 𝑥]𝜒
7 alrimii.4 . . 3 𝑦𝜒
8 sbceq2a 3781 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜒𝜒))
96, 7, 8cbvalv1 2352 . 2 (∀𝑦[𝑦 / 𝑥]𝜒 ↔ ∀𝑥𝜒)
105, 9sylib 219 1 (𝜑 → ∀𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wnf 1775  [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-sbc 3770
This theorem is referenced by: (None)
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