Theorem e22 45502

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 45467	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

This proof depends on syntax axioms:   → wi 4  (   wvd2 45408

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-vd2 45409

This theorem is used by:  e22an  45503  e02  45528  e12  45554  e20  45557  e21  45560  sspwtr  45651  pwtrVD  45654  pwtrrVD  45655  elex22VD  45669  tpid3gVD  45672  en3lplem2VD  45674  imbi12VD  45703  truniALTVD  45708  trintALTVD  45710  onfrALTlem3VD  45717  onfrALTlem2VD  45719  ax6e2eqVD  45737  ax6e2ndeqVD  45739  sb5ALTVD  45743