Theorem syli 39

Description: Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)

Hypotheses

Ref	Expression
syli.1	⊢ (𝜓 → (𝜑 → 𝜒))
syli.2	⊢ (𝜒 → (𝜑 → 𝜃))

Assertion

Ref	Expression
syli	⊢ (𝜓 → (𝜑 → 𝜃))

Proof of Theorem syli

Step	Hyp	Ref	Expression
1		syli.1	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
2		syli.2	. . 3 ⊢ (𝜒 → (𝜑 → 𝜃))
3	2	com12 32	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
4	1, 3	sylcom 30	1 ⊢ (𝜓 → (𝜑 → 𝜃))

Syntax hints:   → wi 4

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

This theorem is referenced by:  ibd  269  bija  380  equvel  2461  2eu6  2657  rgen2a  3371  rexraleqim  3647  elreldm  5946  tz6.12cOLD  6933  onminex  7822  rntpos  8264  smores  8392  seqomlem2  8491  f1domg  9012  php  9247  phpOLD  9259  fodomnum  10097  carduniima  10136  cardmin  10604  negn0  11692  sqrmo  15290  isch3  31260  cgrtriv  36003  wl-lem-moexsb  37569  grpomndo  37882  elghomlem2OLD  37893  tz6.12c-afv2  47254