Theorem e211 45426

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e211.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e211.2	⊢ (   𝜑   ▶   𝜃   )
e211.3	⊢ (   𝜑   ▶   𝜏   )
e211.4	⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))

Assertion

Ref	Expression
e211	⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Proof of Theorem e211

Step	Hyp	Ref	Expression
1		e211.1	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2		e211.2	. . 3 ⊢ (   𝜑   ▶   𝜃   )
3	2	vd12 45369	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
4		e211.3	. . 3 ⊢ (   𝜑   ▶   𝜏   )
5	4	vd12 45369	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
6		e211.4	. 2 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))
7	1, 3, 5, 6	e222 45405	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45338  (   wvd2 45346

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-an 402  df-vd1 45339  df-vd2 45347

This theorem is used by:  e210  45428  e201  45430