Theorem e21 42350

Description: A virtual deduction elimination rule (see syl6ci 71). (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 42220	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
4		e21.3	. 2 ⊢ (𝜒 → (𝜃 → 𝜏))
5	1, 3, 4	e22 42291	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Syntax hints:   → wi 4  (   wvd1 42189  (   wvd2 42197

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-vd2 42198

This theorem is referenced by:  e21an  42351  en3lplem1VD  42463  exbiriVD  42474  syl5impVD  42483  sbcim2gVD  42495  onfrALTlem3VD  42507  onfrALTlem2VD  42509  hbimpgVD  42524  ax6e2eqVD  42527  vk15.4jVD  42534