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  1673  sbequ2  2284  pssnn  9192  kmlem1  10236  brdom5  10615  brdom4  10616  axpowndlem2  10690  naim1  37301  naim2  37302  meran1  37323  bj-gl4  37589  bj-wnf1  37745  rp-fakeanorass  44651  fiinfi  44711  axc11next  45527