Theorem alrimii 38623

Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
alrimii.1	⊢ Ⅎ𝑦𝜑
alrimii.2	⊢ (𝜑 → 𝜓)
alrimii.3	⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)
alrimii.4	⊢ Ⅎ𝑦𝜒

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

Step	Hyp	Ref	Expression
1		alrimii.1	. . 3 ⊢ Ⅎ𝑦𝜑
2		alrimii.2	. . . 4 ⊢ (𝜑 → 𝜓)
3		alrimii.3	. . . 4 ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)
4	2, 3	sylibr 236	. . 3 ⊢ (𝜑 → [𝑦 / 𝑥]𝜒)
5	1, 4	alrimi 2250	. 2 ⊢ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜒)
6		nfsbc1v 3766	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜒
7		alrimii.4	. . 3 ⊢ Ⅎ𝑦𝜒
8		sbceq2a 3758	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜒 ↔ 𝜒))
9	6, 7, 8	cbvalv1 2374	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜒 ↔ ∀𝑥𝜒)
10	5, 9	sylib 220	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560  Ⅎwnf 1805  [wsbc 3746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-sbc 3747

This theorem is referenced by: (None)