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Mirrors > Home > ILE Home > Th. List > sneqrg | GIF version |
Description: Closed form of sneqr 3687. (Contributed by Scott Fenton, 1-Apr-2011.) |
Ref | Expression |
---|---|
sneqrg | ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3538 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | eqeq1d 2148 | . . 3 ⊢ (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵})) |
3 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
4 | 2, 3 | imbi12d 233 | . 2 ⊢ (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵))) |
5 | vex 2689 | . . 3 ⊢ 𝑥 ∈ V | |
6 | 5 | sneqr 3687 | . 2 ⊢ ({𝑥} = {𝐵} → 𝑥 = 𝐵) |
7 | 4, 6 | vtoclg 2746 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 {csn 3527 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sn 3533 |
This theorem is referenced by: sneqbg 3690 |
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