Theorem idandthmr-P4.23b 447

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

Hypothesis

Ref	Expression
idandthmr-P4.23b.1	⊢ 𝜑

Assertion

Ref	Expression
idandthmr-P4.23b	⊢ ((𝜓 ∧ 𝜑) ↔ 𝜓)

Proof of Theorem idandthmr-P4.23b

Step	Hyp	Ref	Expression
1		idandtruer-P4.19b 439	. 2 ⊢ ((𝜓 ∧ ⊤) ↔ 𝜓)
2		idandthmr-P4.23b.1	. . . . . 6 ⊢ 𝜑
3	2	thmeqtrue-P4.21a 442	. . . . 5 ⊢ (𝜑 ↔ ⊤)
4	3	subandr-P3.42b.RC 342	. . . 4 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ ⊤))
5	4	subbil-P3.41a.RC 333	. . 3 ⊢ (((𝜓 ∧ 𝜑) ↔ 𝜓) ↔ ((𝜓 ∧ ⊤) ↔ 𝜓))
6	5	rcp-NDBIER0 241	. 2 ⊢ (((𝜓 ∧ ⊤) ↔ 𝜓) → ((𝜓 ∧ 𝜑) ↔ 𝜓))
7	1, 6	rcp-NDIME0 228	1 ⊢ ((𝜓 ∧ 𝜑) ↔ 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: (None)