Theorem syl3anl1 1420

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)

Hypotheses

Ref	Expression
syl3anl1.1	⊢ (𝜑 → 𝜓)
syl3anl1.2	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl3anl1	⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl1

Step	Hyp	Ref	Expression
1		syl3anl1.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	3anim1i 1158	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃))
3		syl3anl1.2	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
4	2, 3	sylan 586	1 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092

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 208  df-an 397  df-3an 1094

This theorem is referenced by:  dif1enlem  9084  suprzcl  12600  latjcom  18404  latmcom  18420  ring1zr  20748  lgsdinn0  27326  revpfxsfxrev  35344  crngohomfo  38373  dalem53  40217