Theorem dsyl-P3.25.RC 262

Description: Inference Form of dsyl-P3.25 261. †

Hypotheses

Ref	Expression
dsyl-P3.25.RC.1	⊢ (𝜑 → 𝜓)
dsyl-P3.25.RC.2	⊢ (𝜓 → 𝜒)
dsyl-P3.25.RC.3	⊢ (𝜒 → 𝜗)

Assertion

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

Proof of Theorem dsyl-P3.25.RC

Step	Hyp	Ref	Expression
1		dsyl-P3.25.RC.1	. . . 4 ⊢ (𝜑 → 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 → 𝜓))
3		dsyl-P3.25.RC.2	. . . 4 ⊢ (𝜓 → 𝜒)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜓 → 𝜒))
5		dsyl-P3.25.RC.3	. . . 4 ⊢ (𝜒 → 𝜗)
6	5	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 → 𝜗))
7	2, 4, 6	dsyl-P3.25 261	. 2 ⊢ (⊤ → (𝜑 → 𝜗))
8	7	ndtruee-P3.18 183	1 ⊢ (𝜑 → 𝜗)

Syntax hints:   → wff-imp 10  ⊤wff-true 153

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  df-true-D2.4 155

This theorem is referenced by:  lemma-L5.04a  667  nfrim-P6  689  trnsvsubw-P6  710  exipsub-P6  720  trnsvsub-P6  763  splitelof-P6-L1  777  dfnfreeint-P7  969  dfpsubv-P7  977  nfrexall-P8  1086  nfrnegconv-P8  1110