Theorem e1a 42136

Description: A Virtual deduction elimination rule. syl 17 is e1a 42136 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 42080	. . 3 ⊢ (𝜑 → 𝜓)
3		e1a.2	. . 3 ⊢ (𝜓 → 𝜒)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → 𝜒)
5	4	dfvd1ir 42082	1 ⊢ (   𝜑   ▶   𝜒   )

Syntax hints:   → wi 4  (   wvd1 42078

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 42079

This theorem is referenced by:  e1bi  42138  e1bir  42139  snelpwrVD  42340  unipwrVD  42341  sstrALT2VD  42343  elex2VD  42347  elex22VD  42348  eqsbc2VD  42349  zfregs2VD  42350  tpid3gVD  42351  en3lplem1VD  42352  en3lpVD  42354  3ornot23VD  42356  3orbi123VD  42359  sbc3orgVD  42360  exbirVD  42362  3impexpVD  42365  3impexpbicomVD  42366  tratrbVD  42370  al2imVD  42371  syl5impVD  42372  ssralv2VD  42375  ordelordALTVD  42376  sbcim2gVD  42384  trsbcVD  42386  truniALTVD  42387  trintALTVD  42389  undif3VD  42391  sbcssgVD  42392  csbingVD  42393  onfrALTlem3VD  42396  simplbi2comtVD  42397  onfrALTlem2VD  42398  onfrALTVD  42400  csbeq2gVD  42401  csbsngVD  42402  csbxpgVD  42403  csbresgVD  42404  csbrngVD  42405  csbima12gALTVD  42406  csbunigVD  42407  csbfv12gALTVD  42408  con5VD  42409  relopabVD  42410  19.41rgVD  42411  2pm13.193VD  42412  hbimpgVD  42413  hbalgVD  42414  hbexgVD  42415  ax6e2eqVD  42416  ax6e2ndVD  42417  ax6e2ndeqVD  42418  2sb5ndVD  42419  2uasbanhVD  42420  e2ebindVD  42421  sb5ALTVD  42422  vk15.4jVD  42423  notnotrALTVD  42424  con3ALTVD  42425