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Theorem alrimii 35389
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 236 . . 3 (𝜑[𝑦 / 𝑥]𝜒)
51, 4alrimi 2206 . 2 (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜒)
6 nfsbc1v 3790 . . 3 𝑥[𝑦 / 𝑥]𝜒
7 alrimii.4 . . 3 𝑦𝜒
8 sbceq2a 3782 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜒𝜒))
96, 7, 8cbvalv1 2355 . 2 (∀𝑦[𝑦 / 𝑥]𝜒 ↔ ∀𝑥𝜒)
105, 9sylib 220 1 (𝜑 → ∀𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1529  wnf 1778  [wsbc 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-sbc 3771
This theorem is referenced by: (None)
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