Theorem syl6an 1479

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  7114  prarloclem5  7831  ltsopr  7927  suplocsrlem  8139  nominpos  9496  ublbneg  9966  wrdsymb0  11285  ccats1pfxeqrex  11435  absle  11802  rexanre  11933  rexico  11934  climshftlemg  12015  serf0  12065  dvds1lem  12516  dvds2lem  12517  lmconst  15210  addcncntoplem  15555  bj-indind  16841