Theorem syl3anl3 1441

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

Hypotheses

Ref	Expression
syl3anl3.1	⊢ (𝜑 → 𝜃)
syl3anl3.2	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl3anl3	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl3

Step	Hyp	Ref	Expression
1		syl3anl3.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	3anim3i 1172	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → (𝜓 ∧ 𝜒 ∧ 𝜃))
3		syl3anl3.2	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
4	2, 3	sylan 592	1 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  revpfxsfxrev  14829  lgsdirnn0  27545  rdgeqoa  38057  lindsadd  38305  atcvreq0  40129  paddasslem16  40650