Theorem imoverim-P4.30-L1 476

Description: Lemma for imoverim-P4.30a . †

Assertion

Ref	Expression
imoverim-P4.30-L1	⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)))

Proof of Theorem imoverim-P4.30-L1

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . . 5 ⊢ ((((𝜑 → 𝜓) → (𝜑 → 𝜒)) ∧ 𝜓) → 𝜓)
2	1	axL1-P3.21 252	. . . 4 ⊢ ((((𝜑 → 𝜓) → (𝜑 → 𝜒)) ∧ 𝜓) → (𝜑 → 𝜓))
3		rcp-NDASM1of2 193	. . . 4 ⊢ ((((𝜑 → 𝜓) → (𝜑 → 𝜒)) ∧ 𝜓) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
4	2, 3	ndime-P3.6 171	. . 3 ⊢ ((((𝜑 → 𝜓) → (𝜑 → 𝜒)) ∧ 𝜓) → (𝜑 → 𝜒))
5	4	rcp-NDIMI2 224	. 2 ⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
6	5	imcomm-P3.27 265	1 ⊢ (((𝜑 → 𝜓) → (𝜑 → 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  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  imoverim-P4.30a  477