Theorem e21 45553

Description: A virtual deduction elimination rule (see syl6ci 72). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e21.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e21.2	⊢ (   𝜑   ▶   𝜃   )
e21.3	⊢ (𝜒 → (𝜃 → 𝜏))

Assertion

Ref	Expression
e21	⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Proof of Theorem e21

Step	Hyp	Ref	Expression
1		e21.1	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2		e21.2	. . 3 ⊢ (   𝜑   ▶   𝜃   )
3	2	vd12 45424	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
4		e21.3	. 2 ⊢ (𝜒 → (𝜃 → 𝜏))
5	1, 3, 4	e22 45495	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

This proof depends on syntax axioms:   → wi 4  (   wvd1 45393  (   wvd2 45401

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-an 402  df-vd1 45394  df-vd2 45402

This theorem is used by:  e21an  45554  en3lplem1VD  45666  exbiriVD  45677  syl5impVD  45686  sbcim2gVD  45698  onfrALTlem3VD  45710  onfrALTlem2VD  45712  hbimpgVD  45727  ax6e2eqVD  45730  vk15.4jVD  45737