Theorem syl-P1.2 34

Description: Syllogism Inference.

Hypotheses

Ref	Expression
syl-P1.2.1	⊢ (𝜑 → 𝜓)
syl-P1.2.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
syl-P1.2	⊢ (𝜑 → 𝜒)

Proof of Theorem syl-P1.2

Step	Hyp	Ref	Expression
1		syl-P1.2.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl-P1.2.2	. . . 4 ⊢ (𝜓 → 𝜒)
3	2	axL1.SH 30	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
4	3	axL2.SH 31	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜒))
5	1, 4	ax-MP 14	1 ⊢ (𝜑 → 𝜒)

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:  dsyl-P1.3  35  rcp-FR2  41  rcp-FR3  43  poe-P1.11a  65  dneg-P1.13a  71  nclav-P1.14  73  trnsp-P1.15a  76  trnsp-P1.15b  78  trnsp-P1.15c  80  bifwd-P2.5a  111  birev-P2.5b  115  simpl-P2.9a  134  simpr-P2.9b  136  orintr-P2.11b  148  orelim-P2.11c  150