Theorem jad 187

Description: Deduction form of ja 186. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Hypotheses

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

Assertion

Ref	Expression
jad	⊢ (𝜑 → ((𝜓 → 𝜒) → 𝜃))

Proof of Theorem jad

Step	Hyp	Ref	Expression
1		jad.1	. . . 4 ⊢ (𝜑 → (¬ 𝜓 → 𝜃))
2	1	com12 32	. . 3 ⊢ (¬ 𝜓 → (𝜑 → 𝜃))
3		jad.2	. . . 4 ⊢ (𝜑 → (𝜒 → 𝜃))
4	3	com12 32	. . 3 ⊢ (𝜒 → (𝜑 → 𝜃))
5	2, 4	ja 186	. 2 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃))
6	5	com12 32	1 ⊢ (𝜑 → ((𝜓 → 𝜒) → 𝜃))

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:  pm2.6  190  pm2.65  192  merco2  1739  wereu2  5674  frpomin  6342  isfin7-2  10391  axpowndlem3  10594  suppssfz  13959  lo1bdd2  15468  pntlem3  27112  hbimtg  34778  arg-ax  35301  onsuct0  35326  ordcmp  35332  poimirlem26  36514  ax12indi  37814  ntrneiiso  42842  hbimpg  43315