Theorem e01 42311

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e01.1	⊢ 𝜑
e01.2	⊢ (   𝜓   ▶   𝜒   )
e01.3	⊢ (𝜑 → (𝜒 → 𝜃))

Assertion

Ref	Expression
e01	⊢ (   𝜓   ▶   𝜃   )

Proof of Theorem e01

Step	Hyp	Ref	Expression
1		e01.1	. . 3 ⊢ 𝜑
2	1	vd01 42217	. 2 ⊢ (   𝜓   ▶   𝜑   )
3		e01.2	. 2 ⊢ (   𝜓   ▶   𝜒   )
4		e01.3	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5	2, 3, 4	e11 42308	1 ⊢ (   𝜓   ▶   𝜃   )

Syntax hints:   → wi 4  (   wvd1 42189

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-vd1 42190

This theorem is referenced by:  e01an  42312  trsspwALT  42438  sspwtr  42441  pwtrVD  42444  pwtrrVD  42445  snssiALTVD  42447  snelpwrVD  42451  sstrALT2VD  42454  suctrALT2VD  42456  3impexpVD  42476  ax6e2eqVD  42527  ax6e2ndVD  42528  2sb5ndVD  42530  vk15.4jVD  42534