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Theorem pm5.55 867
Description: Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
Assertion
Ref Expression
pm5.55 (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ))

Proof of Theorem pm5.55
StepHypRef Expression
1 biort 866 . . . . 5 (φ → (φ ↔ (φ ψ)))
21bicomd 192 . . . 4 (φ → ((φ ψ) ↔ φ))
3 biorf 394 . . . . 5 φ → (ψ ↔ (φ ψ)))
43bicomd 192 . . . 4 φ → ((φ ψ) ↔ ψ))
52, 4nsyl4 134 . . 3 (¬ ((φ ψ) ↔ ψ) → ((φ ψ) ↔ φ))
65con1i 121 . 2 (¬ ((φ ψ) ↔ φ) → ((φ ψ) ↔ ψ))
76orri 365 1 (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 176   wo 357
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
This theorem is referenced by: (None)
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