Theorem e12 43470

Description: A virtual deduction elimination rule (see sylsyld 61). (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e12.1	⊢ (   𝜑   ▶   𝜓   )
e12.2	⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
e12.3	⊢ (𝜓 → (𝜃 → 𝜏))

Assertion

Ref	Expression
e12	⊢ (   𝜑   ,   𝜒   ▶   𝜏   )

Proof of Theorem e12

Step	Hyp	Ref	Expression
1		e12.1	. . 3 ⊢ (   𝜑   ▶   𝜓   )
2	1	vd12 43346	. 2 ⊢ (   𝜑   ,   𝜒   ▶   𝜓   )
3		e12.2	. 2 ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )
4		e12.3	. 2 ⊢ (𝜓 → (𝜃 → 𝜏))
5	2, 3, 4	e22 43417	1 ⊢ (   𝜑   ,   𝜒   ▶   𝜏   )

Syntax hints:   → wi 4  (   wvd1 43315  (   wvd2 43323

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 43316  df-vd2 43324

This theorem is referenced by:  e12an  43471  trsspwALT  43564  sspwtr  43567  pwtrVD  43570  snssiALTVD  43573  elex2VD  43584  elex22VD  43585  eqsbc2VD  43586  en3lplem1VD  43589  3ornot23VD  43593  orbi1rVD  43594  19.21a3con13vVD  43598  exbirVD  43599  tratrbVD  43607  ssralv2VD  43612  sbcim2gVD  43621  sbcbiVD  43622  relopabVD  43647  19.41rgVD  43648  ax6e2eqVD  43653  ax6e2ndeqVD  43655  vk15.4jVD  43660  con3ALTVD  43662