Theorem dsyl-P1.3 35

Description: Double Syllogism Inference.

Hypotheses

Ref	Expression
dsyl-P1.3.1	⊢ (𝜑 → 𝜓)
dsyl-P1.3.2	⊢ (𝜓 → 𝜒)
dsyl-P1.3.3	⊢ (𝜒 → 𝜗)

Assertion

Ref	Expression
dsyl-P1.3	⊢ (𝜑 → 𝜗)

Proof of Theorem dsyl-P1.3

Step	Hyp	Ref	Expression
1		dsyl-P1.3.1	. . 3 ⊢ (𝜑 → 𝜓)
2		dsyl-P1.3.2	. . 3 ⊢ (𝜓 → 𝜒)
3	1, 2	syl-P1.2 34	. 2 ⊢ (𝜑 → 𝜒)
4		dsyl-P1.3.3	. 2 ⊢ (𝜒 → 𝜗)
5	3, 4	syl-P1.2 34	1 ⊢ (𝜑 → 𝜗)

Syntax hints:   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-MP 14

This theorem is referenced by:  orintl-P2.11a  146