Theorem syldanl 601

Description: A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
syldanl	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Proof of Theorem syldanl

Step	Hyp	Ref	Expression
1		syldanl.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 412	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imdistani 568	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
4		syldanl.2	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
5	3, 4	sylan 579	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396

This theorem is referenced by:  sylanl2  678  oen0  8582  oeordsuc  8590  erth  8749  phplem2  9205  lo1bdd2  15466  grplmulf1o  18934  grplactcnv  18963  trust  24058  efrlim  26820  efrlimOLD  26821  fedgmullem2  33197  submateq  33281  heibor1lem  37171  idlnegcl  37384  igenmin  37426  eqvrelth  37975  sticksstones22  41481  binomcxplemnotnn0  43629  vonioolem1  45906  vonicclem1  45909  smfsuplem1  46037  smflimsuplem4  46049