Theorem e22 41377

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e22.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e22.2	⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
e22.3	⊢ (𝜒 → (𝜃 → 𝜏))

Assertion

Ref	Expression
e22	⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Proof of Theorem e22

Step	Hyp	Ref	Expression
1		e22.1	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2		e22.2	. 2 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )
3		e22.3	. . 3 ⊢ (𝜒 → (𝜃 → 𝜏))
4	3	a1i 11	. 2 ⊢ (𝜒 → (𝜒 → (𝜃 → 𝜏)))
5	1, 1, 2, 4	e222 41342	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Syntax hints:   → wi 4  (   wvd2 41283

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 210  df-an 400  df-vd2 41284

This theorem is referenced by:  e22an  41378  e02  41403  e12  41430  e20  41433  e21  41436  sspwtr  41527  pwtrVD  41530  pwtrrVD  41531  elex22VD  41545  tpid3gVD  41548  en3lplem2VD  41550  imbi12VD  41579  truniALTVD  41584  trintALTVD  41586  onfrALTlem3VD  41593  onfrALTlem2VD  41595  ax6e2eqVD  41613  ax6e2ndeqVD  41615  sb5ALTVD  41619