Theorem e22 45122

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 45087	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Syntax hints:   → wi 4  (   wvd2 45028

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 208  df-an 397  df-vd2 45029

This theorem is referenced by:  e22an  45123  e02  45148  e12  45174  e20  45177  e21  45180  sspwtr  45271  pwtrVD  45274  pwtrrVD  45275  elex22VD  45289  tpid3gVD  45292  en3lplem2VD  45294  imbi12VD  45323  truniALTVD  45328  trintALTVD  45330  onfrALTlem3VD  45337  onfrALTlem2VD  45339  ax6e2eqVD  45357  ax6e2ndeqVD  45359  sb5ALTVD  45363