Theorem imoverand-P4.29a 472

Description: '→' Left Distributes Over '∧'. †

Assertion

Ref	Expression
imoverand-P4.29a	⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Proof of Theorem imoverand-P4.29a

Step	Hyp	Ref	Expression
1		sepimandr-P4.9a.CL 408	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
2		joinimandres-P4.8b.CL 402	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → ((𝜑 ∧ 𝜑) → (𝜓 ∧ 𝜒)))
3		idempotand-P4.25a 450	. . . . 5 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
4	3	subiml-P3.40a.RC 326	. . . 4 ⊢ (((𝜑 ∧ 𝜑) → (𝜓 ∧ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))
5	4	rcp-NDBIEF0 240	. . 3 ⊢ (((𝜑 ∧ 𝜑) → (𝜓 ∧ 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
6	2, 5	syl-P3.24.RC 260	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
7	1, 6	rcp-NDBII0 239	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∧ 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

This theorem is referenced by:  imoverbi-P4.30-L2  478