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Theorem pm5.7dc 860
Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 859. (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.7dc (DECID χ → (((φ χ) ↔ (ψ χ)) ↔ (χ (φψ))))

Proof of Theorem pm5.7dc
StepHypRef Expression
1 orbididc 859 . 2 (DECID χ → ((χ (φψ)) ↔ ((χ φ) ↔ (χ ψ))))
2 orcom 646 . . 3 ((χ φ) ↔ (φ χ))
3 orcom 646 . . 3 ((χ ψ) ↔ (ψ χ))
42, 3bibi12i 218 . 2 (((χ φ) ↔ (χ ψ)) ↔ ((φ χ) ↔ (ψ χ)))
51, 4syl6rbb 186 1 (DECID χ → (((φ χ) ↔ (ψ χ)) ↔ (χ (φψ))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wo 628  DECID wdc 741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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