|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > snjust | Structured version Visualization version GIF version | ||
| Description: Soundness justification theorem for df-sn 4626. (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 2740 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴)) | |
| 2 | 1 | cbvabv 2811 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑧 ∣ 𝑧 = 𝐴} | 
| 3 | eqeq1 2740 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴)) | |
| 4 | 3 | cbvabv 2811 | . 2 ⊢ {𝑧 ∣ 𝑧 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | 
| 5 | 2, 4 | eqtri 2764 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |