Theorem sylcom 28

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)

Hypotheses

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

Assertion

Ref	Expression
sylcom	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem sylcom

Step	Hyp	Ref	Expression
1		sylcom.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		sylcom.2	. . 3 ⊢ (𝜓 → (𝜒 → 𝜃))
3	2	a2i 11	. 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → 𝜃))
4	1, 3	syl 14	1 ⊢ (𝜑 → (𝜓 → 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:  syl5com  29  syl6  33  syli  37  mpbidi  151  stdcn  848  con4biddc  858  jaddc  865  con1biddc  877  necon4addc  2430  necon4bddc  2431  necon4ddc  2432  necon1addc  2436  necon1bddc  2437  dmcosseq  4916  iss  4971  funopg  5269  snon0  6966  metrest  14483