Theorem e01 45865

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 45771	. 2 ⊢ (   𝜓   ▶   𝜑   )
3		e01.2	. 2 ⊢ (   𝜓   ▶   𝜒   )
4		e01.3	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5	2, 3, 4	e11 45862	1 ⊢ (   𝜓   ▶   𝜃   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-vd1 45744

This theorem is used by:  e01an  45866  trsspwALT  45991  sspwtr  45994  pwtrVD  45997  pwtrrVD  45998  snssiALTVD  46000  snelpwrVD  46004  sstrALT2VD  46007  suctrALT2VD  46009  3impexpVD  46029  ax6e2eqVD  46080  ax6e2ndVD  46081  2sb5ndVD  46083  vk15.4jVD  46087