Theorem sylt-P1.9.AC.2SH 62

Description: Deductive Form of sylt-P1.9 61.

Hypotheses

Ref	Expression
sylt-P1.9.AC.2SH.1	⊢ (𝛾 → (𝜑 → 𝜓))
sylt-P1.9.AC.2SH.2	⊢ (𝛾 → (𝜓 → 𝜒))

Assertion

Ref	Expression
sylt-P1.9.AC.2SH	⊢ (𝛾 → (𝜑 → 𝜒))

Proof of Theorem sylt-P1.9.AC.2SH

Step	Hyp	Ref	Expression
1		sylt-P1.9.AC.2SH.2	. 2 ⊢ (𝛾 → (𝜓 → 𝜒))
2		sylt-P1.9.AC.2SH.1	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
3		sylt-P1.9 61	. . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
4	3	axL1.SH 30	. . . . 5 ⊢ (𝛾 → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
5	4	rcp-FR1.SH 40	. . . 4 ⊢ ((𝛾 → (𝜑 → 𝜓)) → (𝛾 → ((𝜓 → 𝜒) → (𝜑 → 𝜒))))
6	2, 5	ax-MP 14	. . 3 ⊢ (𝛾 → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
7	6	rcp-FR1.SH 40	. 2 ⊢ ((𝛾 → (𝜓 → 𝜒)) → (𝛾 → (𝜑 → 𝜒)))
8	1, 7	ax-MP 14	1 ⊢ (𝛾 → (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:  orelim-P2.11c  150