Theorem el021old 42321

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

Hypotheses

Ref	Expression
el021old.1	⊢ 𝜑
el021old.2	⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜃   )
el021old.3	⊢ ((𝜑 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
el021old	⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜏   )

Proof of Theorem el021old

Step	Hyp	Ref	Expression
1		el021old.1	. . 3 ⊢ 𝜑
2		el021old.2	. . . 4 ⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜃   )
3	2	dfvd2ani 42203	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4		el021old.3	. . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
5	1, 3, 4	sylancr 587	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
6	5	dfvd2anir 42204	1 ⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜏   )

Syntax hints:   → wi 4   ∧ wa 396  (   wvd1 42189  (   wvhc2 42200

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-vhc2 42201

This theorem is referenced by:  sspwimpcfVD  42541  suctrALTcfVD  42543