Theorem rnlem 971

Description: Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rnlem	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))

Proof of Theorem rnlem

Step	Hyp	Ref	Expression
1		an4 581	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
2	1	biimpi 119	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
3		an42 582	. . . 4 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
4	3	biimpri 132	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)))
5	2, 4	jca 304	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))
6	3	biimpi 119	. . 3 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
7	6	adantl 275	. 2 ⊢ ((((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))) → ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
8	5, 7	impbii 125	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)