Theorem pclem6 1305

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

Step	Hyp	Ref	Expression
1		bi1 116	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → (𝜓 ∧ ¬ 𝜑)))
2		pm3.4 326	. . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝜑) → (𝜓 → ¬ 𝜑))
3	2	com12 30	. . . . 5 ⊢ (𝜓 → ((𝜓 ∧ ¬ 𝜑) → ¬ 𝜑))
4	1, 3	syl9r 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))
7	5, 6	syl9 71	. . . 4 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (¬ 𝜑 → 𝜑)))
8	4, 7	impbidd 125	. . 3 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 ↔ ¬ 𝜑)))
9		pm5.19 654	. . . 4 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
10	9	pm2.21i 607	. . 3 ⊢ ((𝜑 ↔ ¬ 𝜑) → ⊥)
11	8, 10	syl6com 35	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜓 → ⊥))
12		dfnot 1302	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
13	11, 12	sylibr 132	1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ⊥wfal 1289

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 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  nalset  3908  pwnss  3933  bj-nalset  10686