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))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  imim1i  64  dedlem0b  1058  meredith  1668  sbequ2  2291  pssnn  9152  kmlem1  10133  brdom5  10512  brdom4  10513  axpowndlem2  10582  naim1  36788  naim2  36789  meran1  36810  bj-gl4  37076  bj-wnf1  37232  rp-fakeanorass  44130  fiinfi  44190  axc11next  45007