Theorem imim12i 62

Description: Inference joining two implications. Inference associated with imim12 105. Its associated inference is 3syl 18. (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 16	. 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → 𝜃))
4	1, 3	syl5 34	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  63  dedlem0b  1041  meredith  1645  sbequ2  2244  sbequ2OLD  2245  pssnn  8913  pssnnOLD  8969  kmlem1  9837  brdom5  10216  brdom4  10217  axpowndlem2  10285  naim1  34505  naim2  34506  meran1  34527  bj-gl4  34704  bj-wnf1  34826  rp-fakeanorass  41018  fiinfi  41069  axc11next  41913