Theorem stoic4a 1408

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 1409 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 1001	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)
3		simp1 981	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜑)
4		simp3 983	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜃)
5		stoic4a.2	. 2 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)
6	2, 3, 4, 5	syl3anc 1216	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 964

This theorem is referenced by: (None)