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  953  pm2.74  976  pm5.71  1029  meredith  1641  tbw-bijust  1698  tbw-negdf  1699  merco1  1713  19.38  1839  19.35  1877  sbi2  2302  dfmoeu  2529  moabs  2536  exmoeu  2574  moanimlem  2611  r19.35  3087  r19.21v  3154  elab3gf  3640  elab3g  3641  dfss2  3921  r19.2zb  4447  ralidmw  4459  ralidm  4463  iununi  5048  asymref2  6066  nelaneqOLD  9494  fsuppmapnn0fiub0  13900  itgeq2  25677  frgrwopreglem4a  30254  meran1  36385  imsym1  36392  bj-ssbid2ALT  36637  wl-moteq  37488  axc5c7  38890  axc5c711  38897  eu6w  42649  rp-fakeimass  43485  nanorxor  44278  axc5c4c711  44374  pm2.43cbi  44492  euoreqb  47093  oppcendc  49003