Theorem e2 43480

Description: A virtual deduction elimination rule. syl6 35 is e2 43480 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 43434	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		e2.2	. . 3 ⊢ (𝜒 → 𝜃)
4	2, 3	syl6 35	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
5	4	dfvd2ir 43435	1 ⊢ (   𝜑   ,   𝜓   ▶   𝜃   )

Syntax hints:   → wi 4  (   wvd2 43426

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 43427

This theorem is referenced by:  e2bi  43481  e2bir  43482  sspwtr  43670  pwtrVD  43673  pwtrrVD  43674  suctrALT2VD  43685  tpid3gVD  43691  en3lplem1VD  43692  3ornot23VD  43696  orbi1rVD  43697  19.21a3con13vVD  43701  tratrbVD  43710  syl5impVD  43712  ssralv2VD  43715  truniALTVD  43727  trintALTVD  43729  onfrALTlem3VD  43736  onfrALTlem2VD  43738  onfrALTlem1VD  43739  relopabVD  43750  19.41rgVD  43751  hbimpgVD  43753  ax6e2eqVD  43756  ax6e2ndeqVD  43758  sb5ALTVD  43762  vk15.4jVD  43763  con3ALTVD  43765