Theorem e22 44669

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 44634	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )

Syntax hints:   → wi 4  (   wvd2 44575

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 207  df-an 396  df-vd2 44576

This theorem is referenced by:  e22an  44670  e02  44695  e12  44722  e20  44725  e21  44728  sspwtr  44819  pwtrVD  44822  pwtrrVD  44823  elex22VD  44837  tpid3gVD  44840  en3lplem2VD  44842  imbi12VD  44871  truniALTVD  44876  trintALTVD  44878  onfrALTlem3VD  44885  onfrALTlem2VD  44887  ax6e2eqVD  44905  ax6e2ndeqVD  44907  sb5ALTVD  44911