Theorem axL2.AC.SH 46

Description: Deductive Form of ax-L2 12.

Hypothesis

Ref	Expression
axL2.AC.SH.1	⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
axL2.AC.SH	⊢ (𝛾 → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Proof of Theorem axL2.AC.SH

Step	Hyp	Ref	Expression
1		axL2.AC.SH.1	. 2 ⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))
2		ax-L2 12	. . . 4 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
3	2	axL1.SH 30	. . 3 ⊢ (𝛾 → ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
4	3	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → (𝜑 → (𝜓 → 𝜒))) → (𝛾 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
5	1, 4	ax-MP 14	1 ⊢ (𝛾 → ((𝜑 → 𝜓) → (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:  imcomm-P1.6  48  imsubr-P1.7a  51