Theorem e212 42271

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

Hypotheses

Ref	Expression
e212.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e212.2	⊢ (   𝜑   ▶   𝜃   )
e212.3	⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
e212.4	⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))

Assertion

Ref	Expression
e212	⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Proof of Theorem e212

Step	Hyp	Ref	Expression
1		e212.1	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2		e212.2	. . 3 ⊢ (   𝜑   ▶   𝜃   )
3	2	vd12 42220	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
4		e212.3	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
5		e212.4	. 2 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂)))
6	1, 3, 4, 5	e222 42256	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜂   )

Syntax hints:   → wi 4  (   wvd1 42189  (   wvd2 42197

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 397  df-vd1 42190  df-vd2 42198

This theorem is referenced by: (None)