Theorem e1a 44057

Description: A Virtual deduction elimination rule. syl 17 is e1a 44057 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e1a.1	⊢ (   𝜑   ▶   𝜓   )
e1a.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
e1a	⊢ (   𝜑   ▶   𝜒   )

Proof of Theorem e1a

Step	Hyp	Ref	Expression
1		e1a.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 44001	. . 3 ⊢ (𝜑 → 𝜓)
3		e1a.2	. . 3 ⊢ (𝜓 → 𝜒)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → 𝜒)
5	4	dfvd1ir 44003	1 ⊢ (   𝜑   ▶   𝜒   )

Syntax hints:   → wi 4  (   wvd1 43999

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-vd1 44000

This theorem is referenced by:  e1bi  44059  e1bir  44060  snelpwrVD  44261  unipwrVD  44262  sstrALT2VD  44264  elex2VD  44268  elex22VD  44269  eqsbc2VD  44270  zfregs2VD  44271  tpid3gVD  44272  en3lplem1VD  44273  en3lpVD  44275  3ornot23VD  44277  3orbi123VD  44280  sbc3orgVD  44281  exbirVD  44283  3impexpVD  44286  3impexpbicomVD  44287  tratrbVD  44291  al2imVD  44292  syl5impVD  44293  ssralv2VD  44296  ordelordALTVD  44297  sbcim2gVD  44305  trsbcVD  44307  truniALTVD  44308  trintALTVD  44310  undif3VD  44312  sbcssgVD  44313  csbingVD  44314  onfrALTlem3VD  44317  simplbi2comtVD  44318  onfrALTlem2VD  44319  onfrALTVD  44321  csbeq2gVD  44322  csbsngVD  44323  csbxpgVD  44324  csbresgVD  44325  csbrngVD  44326  csbima12gALTVD  44327  csbunigVD  44328  csbfv12gALTVD  44329  con5VD  44330  relopabVD  44331  19.41rgVD  44332  2pm13.193VD  44333  hbimpgVD  44334  hbalgVD  44335  hbexgVD  44336  ax6e2eqVD  44337  ax6e2ndVD  44338  ax6e2ndeqVD  44339  2sb5ndVD  44340  2uasbanhVD  44341  e2ebindVD  44342  sb5ALTVD  44343  vk15.4jVD  44344  notnotrALTVD  44345  con3ALTVD  44346