Theorem imim12i 63

Description: Inference joining two implications. Inference associated with imim12 106. Its associated inference is 3syl 19. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.)

Hypotheses

Ref	Expression
imim12i.1	⊢ (𝜑 → 𝜓)
imim12i.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
imim12i	⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃))

Proof of Theorem imim12i

Step	Hyp	Ref	Expression
1		imim12i.1	. 2 ⊢ (𝜑 → 𝜓)
2		imim12i.2	. . 3 ⊢ (𝜒 → 𝜃)
3	2	imim2i 17	. 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → 𝜃))
4	1, 3	syl5 35	1 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃))

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:  imim1i  64  dedlem0b  1060  meredith  1674  sbequ2  2285  pssnn  9184  kmlem1  10229  brdom5  10608  brdom4  10609  axpowndlem2  10683  naim1  37177  naim2  37178  meran1  37199  bj-gl4  37465  bj-wnf1  37621  rp-fakeanorass  44513  fiinfi  44573  axc11next  45389