Theorem e122 42226

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

Hypotheses

Ref	Expression
e122.1	⊢ (   𝜑   ▶   𝜓   )
e122.2	⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
e122.3	⊢ (   𝜑   ,   𝜒   ▶   𝜏   )
e122.4	⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂)))

Assertion

Ref	Expression
e122	⊢ (   𝜑   ,   𝜒   ▶   𝜂   )

Proof of Theorem e122

Step	Hyp	Ref	Expression
1		e122.1	. . 3 ⊢ (   𝜑   ▶   𝜓   )
2	1	vd12 42173	. 2 ⊢ (   𝜑   ,   𝜒   ▶   𝜓   )
3		e122.2	. 2 ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
4		e122.3	. 2 ⊢ (   𝜑   ,   𝜒   ▶   𝜏   )
5		e122.4	. 2 ⊢ (𝜓 → (𝜃 → (𝜏 → 𝜂)))
6	2, 3, 4, 5	e222 42209	1 ⊢ (   𝜑   ,   𝜒   ▶   𝜂   )

Syntax hints:   → wi 4  (   wvd1 42142  (   wvd2 42150

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-an 396  df-vd1 42143  df-vd2 42151

This theorem is referenced by:  tratrbVD  42434