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 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 580	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 207  df-an 396

This theorem is referenced by:  sylanl2  681  oen0  8511  oeordsuc  8519  erth  8686  phplem2  9129  lo1bdd2  15449  grplmulf1o  18910  grplactcnv  18940  trust  24133  efrlim  26895  efrlimOLD  26896  fedgmullem2  33602  submateq  33775  heibor1lem  37788  idlnegcl  38001  igenmin  38043  eqvrelth  38587  sticksstones22  42141  binomcxplemnotnn0  44329  vonioolem1  46662  vonicclem1  46665  smfsuplem1  46793  smflimsuplem4  46805