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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 4492 | . . . . . 6 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌) | |
2 | nnel 3099 | . . . . . 6 ⊢ (¬ 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌}) | |
3 | nne 2988 | . . . . . 6 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ 𝑥 = 𝑌) | |
4 | 1, 2, 3 | 3bitr4ri 305 | . . . . 5 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ ¬ 𝑥 ∉ {𝑌}) |
5 | 4 | con4bii 322 | . . . 4 ⊢ (𝑥 ≠ 𝑌 ↔ 𝑥 ∉ {𝑌}) |
6 | 5 | imbi1i 351 | . . 3 ⊢ ((𝑥 ≠ 𝑌 → 𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑)) |
7 | 6 | ralbii 3132 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑)) |
8 | raldifb 4046 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑) | |
9 | 7, 8 | bitri 276 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∉ wnel 3090 ∀wral 3105 ∖ cdif 3860 {csn 4476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-v 3439 df-dif 3866 df-sn 4477 |
This theorem is referenced by: dff14b 6899 |
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