Theorem e1a 44625

Description: A Virtual deduction elimination rule. syl 17 is e1a 44625 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 44569	. . 3 ⊢ (𝜑 → 𝜓)
3		e1a.2	. . 3 ⊢ (𝜓 → 𝜒)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → 𝜒)
5	4	dfvd1ir 44571	1 ⊢ (   𝜑   ▶   𝜒   )

Syntax hints:   → wi 4  (   wvd1 44567

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-vd1 44568

This theorem is referenced by:  e1bi  44627  e1bir  44628  snelpwrVD  44829  unipwrVD  44830  sstrALT2VD  44832  elex2VD  44836  elex22VD  44837  eqsbc2VD  44838  zfregs2VD  44839  tpid3gVD  44840  en3lplem1VD  44841  en3lpVD  44843  3ornot23VD  44845  3orbi123VD  44848  sbc3orgVD  44849  exbirVD  44851  3impexpVD  44854  3impexpbicomVD  44855  tratrbVD  44859  al2imVD  44860  syl5impVD  44861  ssralv2VD  44864  ordelordALTVD  44865  sbcim2gVD  44873  trsbcVD  44875  truniALTVD  44876  trintALTVD  44878  undif3VD  44880  sbcssgVD  44881  csbingVD  44882  onfrALTlem3VD  44885  simplbi2comtVD  44886  onfrALTlem2VD  44887  onfrALTVD  44889  csbeq2gVD  44890  csbsngVD  44891  csbxpgVD  44892  csbresgVD  44893  csbrngVD  44894  csbima12gALTVD  44895  csbunigVD  44896  csbfv12gALTVD  44897  con5VD  44898  relopabVD  44899  19.41rgVD  44900  2pm13.193VD  44901  hbimpgVD  44902  hbalgVD  44903  hbexgVD  44904  ax6e2eqVD  44905  ax6e2ndVD  44906  ax6e2ndeqVD  44907  2sb5ndVD  44908  2uasbanhVD  44909  e2ebindVD  44910  sb5ALTVD  44911  vk15.4jVD  44912  notnotrALTVD  44913  con3ALTVD  44914