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| Mirrors > Home > MPE Home > Th. List > raldifsnb | Structured version Visualization version GIF version | ||
| Description: Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.) | 
| Ref | Expression | 
|---|---|
| raldifsnb | ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | velsn 4642 | . . . . . 6 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌) | |
| 2 | nnel 3056 | . . . . . 6 ⊢ (¬ 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌}) | |
| 3 | nne 2944 | . . . . . 6 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ 𝑥 = 𝑌) | |
| 4 | 1, 2, 3 | 3bitr4ri 304 | . . . . 5 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ ¬ 𝑥 ∉ {𝑌}) | 
| 5 | 4 | con4bii 321 | . . . 4 ⊢ (𝑥 ≠ 𝑌 ↔ 𝑥 ∉ {𝑌}) | 
| 6 | 5 | imbi1i 349 | . . 3 ⊢ ((𝑥 ≠ 𝑌 → 𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑)) | 
| 7 | 6 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑)) | 
| 8 | raldifb 4149 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑) | |
| 9 | 7, 8 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∀wral 3061 ∖ cdif 3948 {csn 4626 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-nel 3047 df-ral 3062 df-v 3482 df-dif 3954 df-sn 4627 | 
| This theorem is referenced by: dff14b 7291 isdomn5 20710 safesnsupfilb 43431 | 
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