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Theorem pclem6 1040
Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.)
Assertion
Ref Expression
pclem6 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Proof of Theorem pclem6
StepHypRef Expression
1 ibar 520 . . 3 (𝜓 → (¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
2 nbbn 374 . . 3 ((¬ 𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) ↔ ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
31, 2sylib 209 . 2 (𝜓 → ¬ (𝜑 ↔ (𝜓 ∧ ¬ 𝜑)))
43con2i 136 1 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  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 198  df-an 385
This theorem is referenced by:  nalset  4990  pwnss  5022  bj-nalset  33108
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