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Theorem rnlem 894
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 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜓𝜃)) ∧ ((𝜑𝜃) ∧ (𝜓𝜒))))

Proof of Theorem rnlem
StepHypRef Expression
1 an4 528 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜓𝜃)))
21biimpi 117 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ∧ (𝜓𝜃)))
3 an42 529 . . . 4 (((𝜑𝜃) ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
43biimpri 128 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜃) ∧ (𝜓𝜒)))
52, 4jca 294 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (((𝜑𝜒) ∧ (𝜓𝜃)) ∧ ((𝜑𝜃) ∧ (𝜓𝜒))))
63biimpi 117 . . 3 (((𝜑𝜃) ∧ (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜒𝜃)))
76adantl 266 . 2 ((((𝜑𝜒) ∧ (𝜓𝜃)) ∧ ((𝜑𝜃) ∧ (𝜓𝜒))) → ((𝜑𝜓) ∧ (𝜒𝜃)))
85, 7impbii 121 1 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜓𝜃)) ∧ ((𝜑𝜃) ∧ (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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