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Theorem anddi 840
Description: Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
anddi (((φ ψ) (χ θ)) ↔ (((φ χ) (φ θ)) ((ψ χ) (ψ θ))))

Proof of Theorem anddi
StepHypRef Expression
1 andir 838 . 2 (((φ ψ) (χ θ)) ↔ ((φ (χ θ)) (ψ (χ θ))))
2 andi 837 . . 3 ((φ (χ θ)) ↔ ((φ χ) (φ θ)))
3 andi 837 . . 3 ((ψ (χ θ)) ↔ ((ψ χ) (ψ θ)))
42, 3orbi12i 507 . 2 (((φ (χ θ)) (ψ (χ θ))) ↔ (((φ χ) (φ θ)) ((ψ χ) (ψ θ))))
51, 4bitri 240 1 (((φ ψ) (χ θ)) ↔ (((φ χ) (φ θ)) ((ψ χ) (ψ θ))))
Colors of variables: wff setvar class
Syntax hints:  wb 176   wo 357   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-or 359  df-an 360
This theorem is referenced by:  funun  5146
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