Theorem e1a 42247

Description: A Virtual deduction elimination rule. syl 17 is e1a 42247 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 42191	. . 3 ⊢ (𝜑 → 𝜓)
3		e1a.2	. . 3 ⊢ (𝜓 → 𝜒)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → 𝜒)
5	4	dfvd1ir 42193	1 ⊢ (   𝜑   ▶   𝜒   )

Syntax hints:   → wi 4  (   wvd1 42189

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 42190

This theorem is referenced by:  e1bi  42249  e1bir  42250  snelpwrVD  42451  unipwrVD  42452  sstrALT2VD  42454  elex2VD  42458  elex22VD  42459  eqsbc2VD  42460  zfregs2VD  42461  tpid3gVD  42462  en3lplem1VD  42463  en3lpVD  42465  3ornot23VD  42467  3orbi123VD  42470  sbc3orgVD  42471  exbirVD  42473  3impexpVD  42476  3impexpbicomVD  42477  tratrbVD  42481  al2imVD  42482  syl5impVD  42483  ssralv2VD  42486  ordelordALTVD  42487  sbcim2gVD  42495  trsbcVD  42497  truniALTVD  42498  trintALTVD  42500  undif3VD  42502  sbcssgVD  42503  csbingVD  42504  onfrALTlem3VD  42507  simplbi2comtVD  42508  onfrALTlem2VD  42509  onfrALTVD  42511  csbeq2gVD  42512  csbsngVD  42513  csbxpgVD  42514  csbresgVD  42515  csbrngVD  42516  csbima12gALTVD  42517  csbunigVD  42518  csbfv12gALTVD  42519  con5VD  42520  relopabVD  42521  19.41rgVD  42522  2pm13.193VD  42523  hbimpgVD  42524  hbalgVD  42525  hbexgVD  42526  ax6e2eqVD  42527  ax6e2ndVD  42528  ax6e2ndeqVD  42529  2sb5ndVD  42530  2uasbanhVD  42531  e2ebindVD  42532  sb5ALTVD  42533  vk15.4jVD  42534  notnotrALTVD  42535  con3ALTVD  42536