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Theorem pm5.55 975
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 (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓))

Proof of Theorem pm5.55
StepHypRef Expression
1 biort 974 . . . . 5 (𝜑 → (𝜑 ↔ (𝜑𝜓)))
21bicomd 213 . . . 4 (𝜑 → ((𝜑𝜓) ↔ 𝜑))
3 biorf 419 . . . . 5 𝜑 → (𝜓 ↔ (𝜑𝜓)))
43bicomd 213 . . . 4 𝜑 → ((𝜑𝜓) ↔ 𝜓))
52, 4nsyl4 156 . . 3 (¬ ((𝜑𝜓) ↔ 𝜓) → ((𝜑𝜓) ↔ 𝜑))
65con1i 144 . 2 (¬ ((𝜑𝜓) ↔ 𝜑) → ((𝜑𝜓) ↔ 𝜓))
76orri 390 1 (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382
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 197  df-or 384
This theorem is referenced by: (None)
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