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Theorem pm5.75 868
Description: Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)
Assertion
Ref Expression
pm5.75 (((χ → ¬ ψ) (φ ↔ (ψ χ))) → ((φ ¬ ψ) ↔ χ))

Proof of Theorem pm5.75
StepHypRef Expression
1 anbi1 439 . . 3 ((φ ↔ (ψ χ)) → ((φ ¬ ψ) ↔ ((ψ χ) ¬ ψ)))
2 orcom 646 . . . . 5 ((ψ χ) ↔ (χ ψ))
32anbi1i 431 . . . 4 (((ψ χ) ¬ ψ) ↔ ((χ ψ) ¬ ψ))
4 pm5.61 707 . . . 4 (((χ ψ) ¬ ψ) ↔ (χ ¬ ψ))
53, 4bitri 173 . . 3 (((ψ χ) ¬ ψ) ↔ (χ ¬ ψ))
61, 5syl6bb 185 . 2 ((φ ↔ (ψ χ)) → ((φ ¬ ψ) ↔ (χ ¬ ψ)))
7 pm4.71 369 . . . 4 ((χ → ¬ ψ) ↔ (χ ↔ (χ ¬ ψ)))
87biimpi 113 . . 3 ((χ → ¬ ψ) → (χ ↔ (χ ¬ ψ)))
98bicomd 129 . 2 ((χ → ¬ ψ) → ((χ ¬ ψ) ↔ χ))
106, 9sylan9bbr 436 1 (((χ → ¬ ψ) (φ ↔ (ψ χ))) → ((φ ¬ ψ) ↔ χ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628
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
This theorem is referenced by: (None)
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