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Mirrors > Home > ILE Home > Th. List > snjust | GIF version |
Description: Soundness justification theorem for df-sn 3597. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
snjust | ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2184 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴)) | |
2 | 1 | cbvabv 2302 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑧 ∣ 𝑧 = 𝐴} |
3 | eqeq1 2184 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴)) | |
4 | 3 | cbvabv 2302 | . 2 ⊢ {𝑧 ∣ 𝑧 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
5 | 2, 4 | eqtri 2198 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 {cab 2163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 |
This theorem is referenced by: (None) |
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