Theorem imim2 59

Description: A closed form of syllogism (see syl 18). Theorem *2.05 of [WhiteheadRussell] p. 100. Its associated inference is imim2i 17. Its associated deduction is imim2d 58. An alternate proof from more basic results is given by ax-1 6 followed by a2d 30. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)

Assertion

Ref	Expression
imim2	⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))

Proof of Theorem imim2

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	imim2d 58	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))

This proof depends on syntax axioms:   → wi 4

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

This theorem is used by:  syldd  73  imim12  106  pm3.34  778  axprlem2  5386  jath  36469  bj-peircestab  37400  bj-spimnfe  37503  19.41rgVD  45869  adh-minim  48040  adh-minimp  48052