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  964  pm3.13dc  965  nfrimi  1571  abnex  4538  vtoclr  4767  funopg  5352  xpider  6761  rerecapb  8998  ixxssixx  10106  difelfzle  10338  txcnp  14953  uspgr2wlkeqi  16088  bj-inf2vnlem1  16357