Theorem import-P3.34a 305

Description: '∧' Importation Law. †

Hypothesis

Ref	Expression
import-P3.34a.1	⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
import-P3.34a	⊢ (𝛾 → ((𝜑 ∧ 𝜓) → 𝜒))

Proof of Theorem import-P3.34a

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜓)) → (𝜑 ∧ 𝜓))
2	1	ndandel-P3.8 173	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜓)) → 𝜓)
3	1	ndander-P3.9 174	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜓)) → 𝜑)
4		import-P3.34a.1	. . . . 5 ⊢ (𝛾 → (𝜑 → (𝜓 → 𝜒)))
5	4	rcp-NDIMP1add1 208	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜓)) → (𝜑 → (𝜓 → 𝜒)))
6	3, 5	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜓)) → (𝜓 → 𝜒))
7	2, 6	ndime-P3.6 171	. 2 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜓)) → 𝜒)
8	7	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((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-P3.34a.RC  306