Theorem stoic4a 1778

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 1779 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 1128	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
3		simp1 1132	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜑)
4		simp3 1134	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜃)
5		stoic4a.2	. 2 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)
6	2, 3, 4, 5	syl3anc 1367	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by:  pnpcan  10927  relogbexp  25360  repnpcan  39229