Theorem idandtruer-P4.19b 439

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

Assertion

Ref	Expression
idandtruer-P4.19b	⊢ ((𝜑 ∧ ⊤) ↔ 𝜑)

Proof of Theorem idandtruer-P4.19b

Step	Hyp	Ref	Expression
1		idandtruel-P4.19a 438	. 2 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)
2		andcomm-P3.35 314	. . . 4 ⊢ ((⊤ ∧ 𝜑) ↔ (𝜑 ∧ ⊤))
3	2	subbil-P3.41a.RC 333	. . 3 ⊢ (((⊤ ∧ 𝜑) ↔ 𝜑) ↔ ((𝜑 ∧ ⊤) ↔ 𝜑))
4	3	rcp-NDBIEF0 240	. 2 ⊢ (((⊤ ∧ 𝜑) ↔ 𝜑) → ((𝜑 ∧ ⊤) ↔ 𝜑))
5	1, 4	rcp-NDIME0 228	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:  idandthmr-P4.23b  447  truthtblfandt-P4.37c  501