Theorem idandtruel-P4.19a 438

Description: '∧' Identity is '⊤' (left). †

Assertion

Ref	Expression
idandtruel-P4.19a	⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)

Proof of Theorem idandtruel-P4.19a

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. 2 ⊢ ((⊤ ∧ 𝜑) → 𝜑)
2		true-P3.44 352	. . . 4 ⊢ ⊤
3	2	rcp-NDIMP0addall 207	. . 3 ⊢ (𝜑 → ⊤)
4		rcp-NDASM1of1 192	. . 3 ⊢ (𝜑 → 𝜑)
5	3, 4	ndandi-P3.7 172	. 2 ⊢ (𝜑 → (⊤ ∧ 𝜑))
6	1, 5	rcp-NDBII0 239	1 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ↔ wff-bi 104   ∧ wff-and 132  ⊤wff-true 153

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:  idandtruer-P4.19b  439  idandthml-P4.23a  446  oroverim-P4.28-L1  465  truthtbltandt-P4.37a  499  truthtbltandf-P4.37b  500