Theorem syl2im 38

Description: Replace two antecedents. Implication-only version of syl2an 289. (Contributed by Wolf Lammen, 14-May-2013.)

Hypotheses

Ref	Expression
syl2im.1	⊢ (𝜑 → 𝜓)
syl2im.2	⊢ (𝜒 → 𝜃)
syl2im.3	⊢ (𝜓 → (𝜃 → 𝜏))

Assertion

Ref	Expression
syl2im	⊢ (𝜑 → (𝜒 → 𝜏))

Proof of Theorem syl2im

Step	Hyp	Ref	Expression
1		syl2im.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl2im.2	. . 3 ⊢ (𝜒 → 𝜃)
3		syl2im.3	. . 3 ⊢ (𝜓 → (𝜃 → 𝜏))
4	2, 3	syl5 32	. 2 ⊢ (𝜓 → (𝜒 → 𝜏))
5	1, 4	syl 14	1 ⊢ (𝜑 → (𝜒 → 𝜏))

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:  syl2imc  39  sylc  62  bi3ant  224  pm3.12dc  971  pm3.13dc  972  nfrimi  1578  abnex  4595  vtoclr  4825  funopg  5414  xpider  6884  rerecapb  9182  ixxssixx  10322  difelfzle  10559  txcnp  15509  uspgr2wlkeqi  16825  bj-inf2vnlem1  17213