Theorem idandthml-P4.23a 446

Description: '∧' Identity is Any Theorem (left). †

Hypothesis

Ref	Expression
idandthml-P4.23a.1	⊢ 𝜑

Assertion

Ref	Expression
idandthml-P4.23a	⊢ ((𝜑 ∧ 𝜓) ↔ 𝜓)

Proof of Theorem idandthml-P4.23a

Step	Hyp	Ref	Expression
1		idandtruel-P4.19a 438	. 2 ⊢ ((⊤ ∧ 𝜓) ↔ 𝜓)
2		idandthml-P4.23a.1	. . . . . 6 ⊢ 𝜑
3	2	thmeqtrue-P4.21a 442	. . . . 5 ⊢ (𝜑 ↔ ⊤)
4	3	subandl-P3.42a.RC 340	. . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ (⊤ ∧ 𝜓))
5	4	subbil-P3.41a.RC 333	. . 3 ⊢ (((𝜑 ∧ 𝜓) ↔ 𝜓) ↔ ((⊤ ∧ 𝜓) ↔ 𝜓))
6	5	rcp-NDBIER0 241	. 2 ⊢ (((⊤ ∧ 𝜓) ↔ 𝜓) → ((𝜑 ∧ 𝜓) ↔ 𝜓))
7	1, 6	rcp-NDIME0 228	1 ⊢ ((𝜑 ∧ 𝜓) ↔ 𝜓)

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:  oroverim-P4.28-L1  465  lemma-L6.07a-L2  771