Theorem stoic4b 1777

Description: Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1776 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)

Hypotheses

Ref	Expression
stoic4b.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
stoic4b.2	⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
stoic4b	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Proof of Theorem stoic4b

Step	Hyp	Ref	Expression
1		stoic4b.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	3adant3 1132	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
3		simp1 1136	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜑)
4		simp2 1137	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜓)
5		simp3 1138	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜃)
6		stoic4b.2	. 2 ⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)
7	2, 3, 4, 5, 6	syl31anc 1374	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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  df-3an 1088

This theorem is referenced by: (None)