|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > eqsb1 | Structured version Visualization version GIF version | ||
| Description: Substitution for the left-hand side in an equality. Class version of equsb3 2102. (Contributed by Rodolfo Medina, 28-Apr-2010.) | 
| Ref | Expression | 
|---|---|
| eqsb1 | ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqeq1 2740 | . 2 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝐴 ↔ 𝑤 = 𝐴)) | |
| 2 | eqeq1 2740 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝐴 ↔ 𝑦 = 𝐴)) | |
| 3 | 1, 2 | sbievw2 2097 | 1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 [wsb 2063 | 
| 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-cleq 2728 | 
| This theorem is referenced by: sbhypf 3543 pm13.183 3665 eqsbc1 3834 | 
| Copyright terms: Public domain | W3C validator |