Theorem rnlem 883

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 520	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
2	1	biimpi 113	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
3		an42 521	. . . 4 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
4	3	biimpri 124	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)))
5	2, 4	jca 290	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))
6	3	biimpi 113	. . 3 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
7	6	adantl 262	. 2 ⊢ ((((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))) → ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
8	5, 7	impbii 117	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))

Syntax hints:   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)