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  855  con4biddc  865  jaddc  872  con1biddc  884  necon4addc  2484  necon4bddc  2485  necon4ddc  2486  necon1addc  2490  necon1bddc  2491  dmcosseq  5031  iss  5086  funopg  5388  snon0  7204  metrest  15388