Theorem pclem6 1369

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		biimp 117	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → (𝜓 ∧ ¬ 𝜑)))
2		pm3.4 331	. . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝜑) → (𝜓 → ¬ 𝜑))
3	2	com12 30	. . . . 5 ⊢ (𝜓 → ((𝜓 ∧ ¬ 𝜑) → ¬ 𝜑))
4	1, 3	syl9r 73	. . . 4 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 → ¬ 𝜑)))
5		ax-ia3 107	. . . . 5 ⊢ (𝜓 → (¬ 𝜑 → (𝜓 ∧ ¬ 𝜑)))
6		biimpr 129	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ((𝜓 ∧ ¬ 𝜑) → 𝜑))
7	5, 6	syl9 72	. . . 4 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (¬ 𝜑 → 𝜑)))
8	4, 7	impbidd 126	. . 3 ⊢ (𝜓 → ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜑 ↔ ¬ 𝜑)))
9		pm5.19 701	. . . 4 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
10	9	pm2.21i 641	. . 3 ⊢ ((𝜑 ↔ ¬ 𝜑) → ⊥)
11	8, 10	syl6com 35	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜓 → ⊥))
12		dfnot 1366	. 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥))
13	11, 12	sylibr 133	1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ⊥wfal 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  nalset  4119  pwnss  4145  bj-nalset  13930