Theorem syl3anl3 1415

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 1155	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → (𝜓 ∧ 𝜒 ∧ 𝜃))
3		syl3anl3.2	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
4	2, 3	sylan 581	1 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜑) ∧ 𝜏) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088

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 398  df-3an 1090

This theorem is referenced by:  lgsdirnn0  26847  revpfxsfxrev  34106  rdgeqoa  36251  lindsadd  36481  atcvreq0  38184  paddasslem16  38706