Theorem syldanl 602

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 413	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	imdistani 569	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
4		syldanl.2	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
5	3, 4	sylan 580	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 396

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 397

This theorem is referenced by:  sylanl2  679  oen0  8588  oeordsuc  8596  erth  8754  phplem2  9210  lo1bdd2  15472  grplmulf1o  18933  grplactcnv  18962  trust  23954  efrlim  26698  fedgmullem2  32991  submateq  33075  heibor1lem  36980  idlnegcl  37193  igenmin  37235  eqvrelth  37784  sticksstones22  41290  binomcxplemnotnn0  43417  vonioolem1  45695  vonicclem1  45698  smfsuplem1  45826  smflimsuplem4  45838