Theorem e01 45441

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 45347	. 2 ⊢ (   𝜓   ▶   𝜑   )
3		e01.2	. 2 ⊢ (   𝜓   ▶   𝜒   )
4		e01.3	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5	2, 3, 4	e11 45438	1 ⊢ (   𝜓   ▶   𝜃   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45319

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 45320

This theorem is used by:  e01an  45442  trsspwALT  45567  sspwtr  45570  pwtrVD  45573  pwtrrVD  45574  snssiALTVD  45576  snelpwrVD  45580  sstrALT2VD  45583  suctrALT2VD  45585  3impexpVD  45605  ax6e2eqVD  45656  ax6e2ndVD  45657  2sb5ndVD  45659  vk15.4jVD  45663