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Mirrors > Home > MPE Home > Th. List > snjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-sn 4632. (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 2739 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴)) | |
2 | 1 | cbvabv 2810 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑧 ∣ 𝑧 = 𝐴} |
3 | eqeq1 2739 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴)) | |
4 | 3 | cbvabv 2810 | . 2 ⊢ {𝑧 ∣ 𝑧 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
5 | 2, 4 | eqtri 2763 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2712 |
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-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 |
This theorem is referenced by: (None) |
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