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Theorem rnlem 931
 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
StepHypRef Expression
1 an4 797 . . . 4 (((φ ψ) (χ θ)) ↔ ((φ χ) (ψ θ)))
21biimpi 186 . . 3 (((φ ψ) (χ θ)) → ((φ χ) (ψ θ)))
3 an42 798 . . . 4 (((φ θ) (ψ χ)) ↔ ((φ ψ) (χ θ)))
43biimpri 197 . . 3 (((φ ψ) (χ θ)) → ((φ θ) (ψ χ)))
52, 4jca 518 . 2 (((φ ψ) (χ θ)) → (((φ χ) (ψ θ)) ((φ θ) (ψ χ))))
63biimpi 186 . . 3 (((φ θ) (ψ χ)) → ((φ ψ) (χ θ)))
76adantl 452 . 2 ((((φ χ) (ψ θ)) ((φ θ) (ψ χ))) → ((φ ψ) (χ θ)))
85, 7impbii 180 1 (((φ ψ) (χ θ)) ↔ (((φ χ) (ψ θ)) ((φ θ) (ψ χ))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-an 360 This theorem is referenced by: (None)
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