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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 4647 | . . . . . 6 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌) | |
2 | nnel 3054 | . . . . . 6 ⊢ (¬ 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌}) | |
3 | nne 2942 | . . . . . 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 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑)) |
8 | raldifb 4159 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∉ wnel 3044 ∀wral 3059 ∖ cdif 3960 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-nel 3045 df-ral 3060 df-v 3480 df-dif 3966 df-sn 4632 |
This theorem is referenced by: dff14b 7291 isdomn5 20727 safesnsupfilb 43408 |
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