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Mirrors > Home > MPE Home > Th. List > raldifsni | Structured version Visualization version GIF version |
Description: Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
Ref | Expression |
---|---|
raldifsni | ⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4751 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵)) | |
2 | 1 | imbi1i 350 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑)) |
3 | impexp 452 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑))) | |
4 | df-ne 2941 | . . . . . 6 ⊢ (𝑥 ≠ 𝐵 ↔ ¬ 𝑥 = 𝐵) | |
5 | 4 | imbi1i 350 | . . . . 5 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑)) |
6 | con34b 316 | . . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑)) | |
7 | 5, 6 | bitr4i 278 | . . . 4 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (𝜑 → 𝑥 = 𝐵)) |
8 | 7 | imbi2i 336 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵))) |
9 | 2, 3, 8 | 3bitri 297 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵))) |
10 | 9 | ralbii2 3089 | 1 ⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ∖ cdif 3911 {csn 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-v 3449 df-dif 3917 df-sn 4591 |
This theorem is referenced by: islindf4 21267 snlindsntor 46642 |
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