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Theorem pclem6 1306
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 116 . . . . 5 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → (𝜓 ∧ ¬ 𝜑)))
2 pm3.4 326 . . . . . 6 ((𝜓 ∧ ¬ 𝜑) → (𝜓 → ¬ 𝜑))
32com12 30 . . . . 5 (𝜓 → ((𝜓 ∧ ¬ 𝜑) → ¬ 𝜑))
41, 3syl9r 72 . . . 4 (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → ¬ 𝜑)))
5 ax-ia3 106 . . . . 5 (𝜓 → (¬ 𝜑 → (𝜓 ∧ ¬ 𝜑)))
6 bi2 128 . . . . 5 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ((𝜓 ∧ ¬ 𝜑) → 𝜑))
75, 6syl9 71 . . . 4 (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (¬ 𝜑𝜑)))
84, 7impbidd 125 . . 3 (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 ↔ ¬ 𝜑)))
9 pm5.19 655 . . . 4 ¬ (𝜑 ↔ ¬ 𝜑)
109pm2.21i 608 . . 3 ((𝜑 ↔ ¬ 𝜑) → ⊥)
118, 10syl6com 35 . 2 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜓 → ⊥))
12 dfnot 1303 . 2 𝜓 ↔ (𝜓 → ⊥))
1311, 12sylibr 132 1 ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wfal 1290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by:  nalset  3916  pwnss  3941  bj-nalset  10844
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