Theorem idempotand-P4.25a 450

Description: Idempotency Law for '∧'. †

Assertion

Ref	Expression
idempotand-P4.25a	⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)

Proof of Theorem idempotand-P4.25a

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜑)
2		rcp-NDASM1of1 192	. . 3 ⊢ (𝜑 → 𝜑)
3	2, 2	ndandi-P3.7 172	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝜑))
4	1, 3	rcp-NDBII0 239	1 ⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ↔ 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:  imoverand-P4.29a  472  imoveror-P4.29-L1  473  truthtblfandf-P4.37d  502  joinimandres3-P4  582  joinimor3-P4  586