Theorem e111 42294

Description: A virtual deduction elimination rule (see syl3c 66). (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 42191	. . . 4 ⊢ (𝜑 → 𝜃)
3		e111.1	. . . . . . 7 ⊢ (   𝜑   ▶   𝜓   )
4	3	in1 42191	. . . . . 6 ⊢ (𝜑 → 𝜓)
5		e111.2	. . . . . . 7 ⊢ (   𝜑   ▶   𝜒   )
6	5	in1 42191	. . . . . 6 ⊢ (𝜑 → 𝜒)
7		e111.4	. . . . . 6 ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))
8	4, 6, 7	syl2im 40	. . . . 5 ⊢ (𝜑 → (𝜑 → (𝜃 → 𝜏)))
9	8	pm2.43i 52	. . . 4 ⊢ (𝜑 → (𝜃 → 𝜏))
10	2, 9	syl5com 31	. . 3 ⊢ (𝜑 → (𝜑 → 𝜏))
11	10	pm2.43i 52	. 2 ⊢ (𝜑 → 𝜏)
12	11	dfvd1ir 42193	1 ⊢ (   𝜑   ▶   𝜏   )

Syntax hints:   → wi 4  (   wvd1 42189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-vd1 42190

This theorem is referenced by:  e110  42296  e101  42298  e011  42300  e100  42302  e010  42304  e001  42306  e11  42308  ordelordALTVD  42487