Theorem tsyl-P4.15 426

Description: Triple Syllogism. †

Hypotheses

Ref	Expression
tsyl-P4.15.1	⊢ (𝛾 → (𝜑 → 𝜓))
tsyl-P4.15.2	⊢ (𝛾 → (𝜓 → 𝜒))
tsyl-P4.15.3	⊢ (𝛾 → (𝜒 → 𝜗))
tsyl-P4.15.4	⊢ (𝛾 → (𝜗 → 𝜏))

Assertion

Ref	Expression
tsyl-P4.15	⊢ (𝛾 → (𝜑 → 𝜏))

Proof of Theorem tsyl-P4.15

Step	Hyp	Ref	Expression
1		tsyl-P4.15.1	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
2		tsyl-P4.15.2	. . 3 ⊢ (𝛾 → (𝜓 → 𝜒))
3		tsyl-P4.15.3	. . 3 ⊢ (𝛾 → (𝜒 → 𝜗))
4	1, 2, 3	dsyl-P3.25 261	. 2 ⊢ (𝛾 → (𝜑 → 𝜗))
5		tsyl-P4.15.4	. 2 ⊢ (𝛾 → (𝜗 → 𝜏))
6	4, 5	syl-P3.24 259	1 ⊢ (𝛾 → (𝜑 → 𝜏))

Syntax hints:   → wff-imp 10

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133

This theorem is referenced by:  tsyl-P4.15.RC  427