Theorem stoic4a 1810

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 1811 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 1150	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
3		simp1 1154	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜑)
4		simp3 1156	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜃)
5		stoic4a.2	. 2 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)
6	2, 3, 4, 5	syl3anc 1398	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

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:  pnpcan  11597  relogbexp  27108  repnpcan  43443