Theorem imoveror-P4.29-L1 473

Description: Lemma for imoveror-P4.29b 474 and imoverorint-P4.29c 475. †

Assertion

Ref	Expression
imoveror-P4.29-L1	⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∨ 𝜒)))

Proof of Theorem imoveror-P4.29-L1

Step	Hyp	Ref	Expression
1		joinimor-P4.8c.CL 405	. 2 ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) → ((𝜑 ∧ 𝜑) → (𝜓 ∨ 𝜒)))
2		idempotand-P4.25a 450	. . . 4 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
3	2	subiml-P3.40a.RC 326	. . 3 ⊢ (((𝜑 ∧ 𝜑) → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (𝜓 ∨ 𝜒)))
4	3	rcp-NDBIEF0 240	. 2 ⊢ (((𝜑 ∧ 𝜑) → (𝜓 ∨ 𝜒)) → (𝜑 → (𝜓 ∨ 𝜒)))
5	1, 4	syl-P3.24.RC 260	1 ⊢ (((𝜑 → 𝜓) ∨ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144

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-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  imoveror-P4.29b  474  imoverorint-P4.29c  475