Theorem e111 45657

Description: A virtual deduction elimination rule (see syl3c 67). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e111.1	⊢ (   𝜑   ▶   𝜓   )
e111.2	⊢ (   𝜑   ▶   𝜒   )
e111.3	⊢ (   𝜑   ▶   𝜃   )
e111.4	⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))

Assertion

Ref	Expression
e111	⊢ (   𝜑   ▶   𝜏   )

Proof of Theorem e111

Step	Hyp	Ref	Expression
1		e111.3	. . . . 5 ⊢ (   𝜑   ▶   𝜃   )
2	1	in1 45554	. . . 4 ⊢ (𝜑 → 𝜃)
3		e111.1	. . . . . . 7 ⊢ (   𝜑   ▶   𝜓   )
4	3	in1 45554	. . . . . 6 ⊢ (𝜑 → 𝜓)
5		e111.2	. . . . . . 7 ⊢ (   𝜑   ▶   𝜒   )
6	5	in1 45554	. . . . . 6 ⊢ (𝜑 → 𝜒)
7		e111.4	. . . . . 6 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
8	4, 6, 7	syl2im 41	. . . . 5 ⊢ (𝜑 → (𝜑 → (𝜃 → 𝜏)))
9	8	pm2.43i 53	. . . 4 ⊢ (𝜑 → (𝜃 → 𝜏))
10	2, 9	syl5com 32	. . 3 ⊢ (𝜑 → (𝜑 → 𝜏))
11	10	pm2.43i 53	. 2 ⊢ (𝜑 → 𝜏)
12	11	dfvd1ir 45556	1 ⊢ (   𝜑   ▶   𝜏   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45552

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-vd1 45553

This theorem is used by:  e110  45659  e101  45661  e011  45663  e100  45665  e010  45667  e001  45669  e11  45671  ordelordALTVD  45849