Theorem syl6an 1454

Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)

Hypotheses

Ref	Expression
syl6an.1	⊢ (𝜑 → 𝜓)
syl6an.2	⊢ (𝜑 → (𝜒 → 𝜃))
syl6an.3	⊢ ((𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl6an	⊢ (𝜑 → (𝜒 → 𝜏))

Proof of Theorem syl6an

Step	Hyp	Ref	Expression
1		syl6an.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
2		syl6an.1	. . 3 ⊢ (𝜑 → 𝜓)
3	1, 2	jctild 316	. 2 ⊢ (𝜑 → (𝜒 → (𝜓 ∧ 𝜃)))
4		syl6an.3	. 2 ⊢ ((𝜓 ∧ 𝜃) → 𝜏)
5	3, 4	syl6 33	1 ⊢ (𝜑 → (𝜒 → 𝜏))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108

This theorem is referenced by:  mapxpen  6952  prarloclem5  7620  ltsopr  7716  suplocsrlem  7928  nominpos  9282  ublbneg  9741  wrdsymb0  11033  absle  11444  rexanre  11575  rexico  11576  climshftlemg  11657  serf0  11707  dvds1lem  12157  dvds2lem  12158  lmconst  14732  addcncntoplem  15077  bj-indind  15942