Theorem alrimii 35557

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 237	. . 3 ⊢ (𝜑 → [𝑦 / 𝑥]𝜒)
5	1, 4	alrimi 2211	. 2 ⊢ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜒)
6		nfsbc1v 3740	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜒
7		alrimii.4	. . 3 ⊢ Ⅎ𝑦𝜒
8		sbceq2a 3732	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜒 ↔ 𝜒))
9	6, 7, 8	cbvalv1 2350	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜒 ↔ ∀𝑥𝜒)
10	5, 9	sylib 221	1 ⊢ (𝜑 → ∀𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-sbc 3721

This theorem is referenced by: (None)