Theorem rnlem 989

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 592	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
2	1	biimpi 120	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
3		an42 593	. . . 4 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
4	3	biimpri 133	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)))
5	2, 4	jca 306	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))
6	3	biimpi 120	. . 3 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
7	6	adantl 277	. 2 ⊢ ((((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))) → ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
8	5, 7	impbii 126	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by: (None)