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Theorem alrimii 33094
 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 223 . . 3 (𝜑[𝑦 / 𝑥]𝜒)
51, 4alrimi 2069 . 2 (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜒)
6 nfsbc1v 3422 . . 3 𝑥[𝑦 / 𝑥]𝜒
7 alrimii.4 . . 3 𝑦𝜒
8 sbceq2a 3414 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜒𝜒))
96, 7, 8cbval 2259 . 2 (∀𝑦[𝑦 / 𝑥]𝜒 ↔ ∀𝑥𝜒)
105, 9sylib 207 1 (𝜑 → ∀𝑥𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-sbc 3403 This theorem is referenced by: (None)
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