Theorem stoic2a 1778

Description: Stoic logic Thema 2 version a. Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as 𝜑, 𝜓⊢ 𝜒; in Metamath we will represent that construct as 𝜑 ∧ 𝜓 → 𝜒. This version a is without the phrase "or both"; see stoic2b 1779 for the version with the phrase "or both". We already have this rule as syldan 590, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

Hypotheses

Ref	Expression
stoic2a.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
stoic2a.2	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
stoic2a	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem stoic2a

Step	Hyp	Ref	Expression
1		stoic2a.1	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2		stoic2a.2	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
3	1, 2	syldan 590	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395

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 206  df-an 396

This theorem is referenced by: (None)