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))

Syntax hints:   → wi 4

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

This theorem is referenced by:  syl2imc  39  sylc  62  bi3ant  224  pm3.12dc  966  pm3.13dc  967  nfrimi  1573  abnex  4546  vtoclr  4776  funopg  5362  xpider  6780  rerecapb  9028  ixxssixx  10142  difelfzle  10374  txcnp  15024  uspgr2wlkeqi  16247  bj-inf2vnlem1  16625