Theorem alrim3con13v 45460

Description: Closed form of alrimi 2249 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 45778 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
alrim3con13v	⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))

Distinct variable groups:   𝜓,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem alrim3con13v

Step	Hyp	Ref	Expression
1		simp1 1154	. . . . 5 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜓)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜓))
3		ax-5 1943	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	2, 3	syl6 36	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜓))
5		simp2 1155	. . . 4 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜑)
6	5	imim1i 64	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜑))
7		simp3 1156	. . . . 5 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜒)
8	7	a1i 11	. . . 4 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜒))
9		ax-5 1943	. . . 4 ⊢ (𝜒 → ∀𝑥𝜒)
10	8, 9	syl6 36	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜒))
11	4, 6, 10	3jcad 1147	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)))
12		19.26-3an 1905	. 2 ⊢ (∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒))
13	11, 12	imbitrrdi 255	1 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  tratrbVD  45787