Theorem tbsyl 34502

Description: The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
tbsyl.1	⊢ (𝜑 → 𝜓)
tbsyl.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
tbsyl	⊢ (𝜑 → 𝜒)

Proof of Theorem tbsyl

Step	Hyp	Ref	Expression
1		tbsyl.2	. 2 ⊢ (𝜓 → 𝜒)
2		tbsyl.1	. . 3 ⊢ (𝜑 → 𝜓)
3		tb-ax1 34499	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒))
5	1, 4	ax-mp 5	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  re1ax2lem  34503  re1ax2  34504