Theorem jad 189

Description: Deduction form of ja 188. (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 33	. . 3 ⊢ (¬ 𝜓 → (𝜑 → 𝜃))
3		jad.2	. . . 4 ⊢ (𝜑 → (𝜒 → 𝜃))
4	3	com12 33	. . 3 ⊢ (𝜒 → (𝜑 → 𝜃))
5	2, 4	ja 188	. 2 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃))
6	5	com12 33	1 ⊢ (𝜑 → ((𝜓 → 𝜒) → 𝜃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  pm2.6  193  pm2.65  195  merco2  1769  wereu2  5648  frpomin  6343  isfin7-2  10474  axpowndlem3  10684  suppssfz  14137  lo1bdd2  15691  pntlem3  27936  hbimtg  36568  arg-ax  37204  onsuct0  37229  ordcmp  37235  poimirlem26  38564  ax12indi  40001  ntrneiiso  45090  hbimpg  45536