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

Step	Hyp	Ref	Expression
1		bi1 111	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → (𝜓 ∧ ¬ 𝜑)))
2		pm3.4 316	. . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝜑) → (𝜓 → ¬ 𝜑))
3	2	com12 27	. . . . 5 ⊢ (𝜓 → ((𝜓 ∧ ¬ 𝜑) → ¬ 𝜑))
4	1, 3	syl9r 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))
7	5, 6	syl9 66	. . . 4 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (¬ 𝜑 → 𝜑)))
8	4, 7	impbidd 118	. . 3 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 ↔ ¬ 𝜑)))
9		pm5.19 622	. . . 4 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
10	9	pm2.21i 575	. . 3 ⊢ ((𝜑 ↔ ¬ 𝜑) → ⊥)
11	8, 10	syl6com 31	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜓 → ⊥))
12		dfnot 1262	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
13	11, 12	sylibr 137	1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

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