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| Description: Substitution applied to the atomic wff with equality. Variant of equsb3 2102. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.) | 
| Ref | Expression | 
|---|---|
| equsb3r | ⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equequ2 2024 | . 2 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤)) | |
| 2 | equequ2 2024 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
| 3 | 1, 2 | sbievw2 2097 | 1 ⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 [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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 | 
| This theorem is referenced by: icheq 47454 | 
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