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Theorem pm5.75OLD 978
Description: Obsolete proof of pm5.75 977 as of 12-Feb-2021. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
pm5.75OLD (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))

Proof of Theorem pm5.75OLD
StepHypRef Expression
1 anbi1 742 . . 3 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓𝜒) ∧ ¬ 𝜓)))
2 orcom 402 . . . . 5 ((𝜓𝜒) ↔ (𝜒𝜓))
32anbi1i 730 . . . 4 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ ((𝜒𝜓) ∧ ¬ 𝜓))
4 pm5.61 748 . . . 4 (((𝜒𝜓) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
53, 4bitri 264 . . 3 (((𝜓𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))
61, 5syl6bb 276 . 2 ((𝜑 ↔ (𝜓𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)))
7 pm4.71 661 . . . 4 ((𝜒 → ¬ 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
87biimpi 206 . . 3 ((𝜒 → ¬ 𝜓) → (𝜒 ↔ (𝜒 ∧ ¬ 𝜓)))
98bicomd 213 . 2 ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒))
106, 9sylan9bbr 736 1 (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384
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 385  df-an 386
This theorem is referenced by: (None)
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