Theorem dsyl-P3.25 261

Description: Double Syllogism. †

Hypotheses

Ref	Expression
dsyl-P3.25.1	⊢ (𝛾 → (𝜑 → 𝜓))
dsyl-P3.25.2	⊢ (𝛾 → (𝜓 → 𝜒))
dsyl-P3.25.3	⊢ (𝛾 → (𝜒 → 𝜗))

Assertion

Ref	Expression
dsyl-P3.25	⊢ (𝛾 → (𝜑 → 𝜗))

Proof of Theorem dsyl-P3.25

Step	Hyp	Ref	Expression
1		dsyl-P3.25.1	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
2		dsyl-P3.25.2	. . 3 ⊢ (𝛾 → (𝜓 → 𝜒))
3	1, 2	syl-P3.24 259	. 2 ⊢ (𝛾 → (𝜑 → 𝜒))
4		dsyl-P3.25.3	. 2 ⊢ (𝛾 → (𝜒 → 𝜗))
5	3, 4	syl-P3.24 259	1 ⊢ (𝛾 → (𝜑 → 𝜗))

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:  dsyl-P3.25.RC  262  tsyl-P4.15  426  qimeqallav-P5-L1  617  qimeqallbv-P5-L1  619  nfrimd-P6  815  lemma-L7.02a-L1  943  lemma-L7.02a  944  nfrnegconv-P8  1110