Theorem export-P2.10b 142

Description: '∧' Exportation Law.

Assertion

Ref	Expression
export-P2.10b	⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))

Proof of Theorem export-P2.10b

Step	Hyp	Ref	Expression
1		cmb-P2.9c 138	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
2	1	axL1.SH 30	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))))
3		ax-L1 11	. . 3 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜓 → ((𝜑 ∧ 𝜓) → 𝜒)))
4	3	axL1.AC.SH 45	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → ((𝜑 ∧ 𝜓) → 𝜒))))
5	2, 4	mpt-P1.8.3AC.2SH 60	1 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132

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:  export-P2.10b.SH  143