Theorem ja 186

Description: Inference joining the antecedents of two premises. For partial converses, see jarri 107 and jarli 126. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.)

Hypotheses

Ref	Expression
ja.1	⊢ (¬ 𝜑 → 𝜒)
ja.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
ja	⊢ ((𝜑 → 𝜓) → 𝜒)

Proof of Theorem ja

Step	Hyp	Ref	Expression
1		ja.2	. . 3 ⊢ (𝜓 → 𝜒)
2	1	imim2i 16	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
3		ja.1	. 2 ⊢ (¬ 𝜑 → 𝜒)
4	2, 3	pm2.61d1 180	1 ⊢ ((𝜑 → 𝜓) → 𝜒)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  jad  187  pm2.01  188  peirce  202  oibabs  954  pm2.74  977  pm5.71  1030  meredith  1643  tbw-bijust  1700  tbw-negdf  1701  merco1  1715  19.38  1841  19.35  1879  sbi2  2309  dfmoeu  2536  moabs  2544  exmoeu  2582  moanimlem  2619  r19.35  3096  r19.21v  3163  elab3gf  3641  elab3g  3642  dfss2  3921  r19.2zb  4455  ralidmw  4471  ralidm  4472  iununi  5056  asymref2  6082  nelaneqOLD  9519  fsuppmapnn0fiub0  13928  itgeq2  25747  frgrwopreglem4a  30397  meran1  36627  imsym1  36634  bj-cbvaw  36884  bj-cbveaw  36886  bj-ssbid2ALT  36908  wl-moteq  37769  axc5c7  39287  axc5c711  39294  eu6w  43034  rp-fakeimass  43868  nanorxor  44661  axc5c4c711  44757  pm2.43cbi  44874  euoreqb  47469  oppcendc  49377