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  3088  r19.35OLD  3089  r19.21v  3158  elab3gf  3651  elab3g  3652  dfss2  3932  r19.2zb  4459  ralidmw  4471  ralidm  4475  iununi  5063  asymref2  6090  nelaneq  9552  fsuppmapnn0fiub0  13958  itgeq2  25679  frgrwopreglem4a  30239  meran1  36399  imsym1  36406  bj-ssbid2ALT  36651  wl-moteq  37502  axc5c7  38904  axc5c711  38911  eu6w  42664  rp-fakeimass  43501  nanorxor  44294  axc5c4c711  44390  pm2.43cbi  44508  euoreqb  47110  oppcendc  49007