Theorem e2 43377

Description: A virtual deduction elimination rule. syl6 35 is e2 43377 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e2.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e2.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
e2	⊢ (   𝜑   ,   𝜓   ▶   𝜃   )

Proof of Theorem e2

Step	Hyp	Ref	Expression
1		e2.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	dfvd2i 43331	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		e2.2	. . 3 ⊢ (𝜒 → 𝜃)
4	2, 3	syl6 35	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
5	4	dfvd2ir 43332	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )

Syntax hints:   → wi 4  (   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-vd2 43324

This theorem is referenced by:  e2bi  43378  e2bir  43379  sspwtr  43567  pwtrVD  43570  pwtrrVD  43571  suctrALT2VD  43582  tpid3gVD  43588  en3lplem1VD  43589  3ornot23VD  43593  orbi1rVD  43594  19.21a3con13vVD  43598  tratrbVD  43607  syl5impVD  43609  ssralv2VD  43612  truniALTVD  43624  trintALTVD  43626  onfrALTlem3VD  43633  onfrALTlem2VD  43635  onfrALTlem1VD  43636  relopabVD  43647  19.41rgVD  43648  hbimpgVD  43650  ax6e2eqVD  43653  ax6e2ndeqVD  43655  sb5ALTVD  43659  vk15.4jVD  43660  con3ALTVD  43662