Theorem alrim3con13v 40860

Description: Closed form of alrimi 2208 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 41179 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 1132	. . . . 5 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜓)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜓))
3		ax-5 1907	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	2, 3	syl6 35	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜓))
5		simp2 1133	. . . 4 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜑)
6	5	imim1i 63	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜑))
7		simp3 1134	. . . . 5 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜒)
8	7	a1i 11	. . . 4 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜒))
9		ax-5 1907	. . . 4 ⊢ (𝜒 → ∀𝑥𝜒)
10	8, 9	syl6 35	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜒))
11	4, 6, 10	3jcad 1125	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)))
12		19.26-3an 1869	. 2 ⊢ (∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒))
13	11, 12	syl6ibr 254	1 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ w3a 1083  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085

This theorem is referenced by:  tratrbVD  41188