Theorem import-P2.10a 140

Description: '∧' Importation Law.

Assertion

Ref	Expression
import-P2.10a	⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → 𝜒))

Proof of Theorem import-P2.10a

Step	Hyp	Ref	Expression
1		simpr-P2.9b 136	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2	1	axL1.SH 30	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → 𝜓))
3		simpl-P2.9a 134	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	3	axL1.SH 30	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → 𝜑))
5		ax-L1 11	. . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → (𝜑 → (𝜓 → 𝜒))))
6	4, 5	mpt-P1.8.2AC.2SH 59	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → (𝜓 → 𝜒)))
7	2, 6	mpt-P1.8.2AC.2SH 59	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:  import-P2.10a.SH  141