Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > pclem6 | GIF version |
Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
Ref | Expression |
---|---|
pclem6 | ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓) |
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 | pm5.19 696 | . . . 4 ⊢ ¬ (𝜑 ↔ ¬ 𝜑) | |
10 | 9 | pm2.21i 636 | . . 3 ⊢ ((𝜑 ↔ ¬ 𝜑) → ⊥) |
11 | 8, 10 | syl6com 35 | . 2 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → (𝜓 → ⊥)) |
12 | dfnot 1353 | . 2 ⊢ (¬ 𝜓 ↔ (𝜓 → ⊥)) | |
13 | 11, 12 | sylibr 133 | 1 ⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ⊥wfal 1340 |
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 604 ax-in2 605 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 |
This theorem is referenced by: nalset 4094 pwnss 4120 bj-nalset 13470 |
Copyright terms: Public domain | W3C validator |