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Mirrors > Home > MPE Home > Th. List > snjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-sn 4211. (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 2655 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴)) | |
2 | 1 | cbvabv 2776 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑧 ∣ 𝑧 = 𝐴} |
3 | eqeq1 2655 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴)) | |
4 | 3 | cbvabv 2776 | . 2 ⊢ {𝑧 ∣ 𝑧 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
5 | 2, 4 | eqtri 2673 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 {cab 2637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 |
This theorem is referenced by: (None) |
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