Theorem jad 173

Description: Deduction form of ja 172. (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 172	. 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  181  pm2.65  183  merco2  1652  nfimd  1812  wereu2  5035  isfin7-2  9101  axpowndlem3  9300  suppssfz  12656  lo1bdd2  14103  pntlem3  25098  hbimtg  30956  arg-ax  31585  onsuct0  31610  ordcmp  31616  poimirlem26  32605  ax12indi  33247  ntrneiiso  37409  hbimpg  37791