Theorem stoic4a 1776

Description: Stoic logic Thema 4 version a. Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic Thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)."

We use 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1777 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

Hypotheses

Ref	Expression
stoic4a.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
stoic4a.2	⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
stoic4a	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Proof of Theorem stoic4a

Step	Hyp	Ref	Expression
1		stoic4a.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	3adant3 1133	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
3		simp1 1137	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜑)
4		simp3 1139	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜃)
5		stoic4a.2	. 2 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)
6	2, 3, 4, 5	syl3anc 1372	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  pnpcan  11555  relogbexp  26849  repnpcan  42415