Theorem syl2im 38

Description: Replace two antecedents. Implication-only version of syl2an 287. (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  223  pm3.12dc  948  pm3.13dc  949  nfrimi  1513  abnex  4425  vtoclr  4652  funopg  5222  xpider  6572  ixxssixx  9838  difelfzle  10069  txcnp  12911  bj-inf2vnlem1  13852