This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem alrim3con13v

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

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

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → 𝜓 )
2 1 a1i ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → 𝜓 ) )
3 ax-5 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
4 2 3 syl6 ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → ∀ 𝑥 𝜓 ) )
5 simp2 ⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → 𝜑 )
6 5 imim1i ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → ∀ 𝑥 𝜑 ) )
7 simp3 ⊢ ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → 𝜒 )
8 7 a1i ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → 𝜒 ) )
9 ax-5 ⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
10 8 9 syl6 ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → ∀ 𝑥 𝜒 ) )
11 4 6 10 3jcad ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → ( ∀ 𝑥 𝜓 ∧ ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜒 ) ) )
12 19.26-3an ⊢ ( ∀ 𝑥 ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) ↔ ( ∀ 𝑥 𝜓 ∧ ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜒 ) )
13 11 12 imbitrrdi ⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜓 ∧ 𝜑 ∧ 𝜒 ) ) )