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Theorem pclem6 1265
 Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
Assertion
Ref Expression
pclem6 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Proof of Theorem pclem6
StepHypRef Expression
1 bi1 111 . . . . 5 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → (𝜓 ∧ ¬ 𝜑)))
2 pm3.4 316 . . . . . 6 ((𝜓 ∧ ¬ 𝜑) → (𝜓 → ¬ 𝜑))
32com12 27 . . . . 5 (𝜓 → ((𝜓 ∧ ¬ 𝜑) → ¬ 𝜑))
41, 3syl9r 67 . . . 4 (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → ¬ 𝜑)))
5 ax-ia3 101 . . . . 5 (𝜓 → (¬ 𝜑 → (𝜓 ∧ ¬ 𝜑)))
6 bi2 121 . . . . 5 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ((𝜓 ∧ ¬ 𝜑) → 𝜑))
75, 6syl9 66 . . . 4 (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (¬ 𝜑𝜑)))
84, 7impbidd 118 . . 3 (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 ↔ ¬ 𝜑)))
9 pm5.19 622 . . . 4 ¬ (𝜑 ↔ ¬ 𝜑)
109pm2.21i 575 . . 3 ((𝜑 ↔ ¬ 𝜑) → ⊥)
118, 10syl6com 31 . 2 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜓 → ⊥))
12 dfnot 1262 . 2 𝜓 ↔ (𝜓 → ⊥))
1311, 12sylibr 137 1 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ⊥wfal 1248 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-in1 544  ax-in2 545 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249 This theorem is referenced by:  nalset  3887  pwnss  3912  bj-nalset  10015
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