Theorem el0321old 42337

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
el0321old.1	⊢ 𝜑
el0321old.2	⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜏   )
el0321old.3	⊢ ((𝜑 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
el0321old	⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜂   )

Proof of Theorem el0321old

Step	Hyp	Ref	Expression
1		el0321old.1	. . 3 ⊢ 𝜑
2		el0321old.2	. . . 4 ⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜏   )
3	2	dfvd3ani 42215	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)
4		el0321old.3	. . 3 ⊢ ((𝜑 ∧ 𝜏) → 𝜂)
5	1, 3, 4	eel0321old 42336	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜂)
6	5	dfvd3anir 42216	1 ⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ wa 396  (   wvd1 42189  (   wvhc3 42208

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-vhc3 42209

This theorem is referenced by:  suctrALTcfVD  42543