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Mirrors > Home > ILE Home > Th. List > equsb3lem | GIF version |
Description: Lemma for equsb3 1874. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
equsb3lem | ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1465 | . 2 ⊢ (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧) | |
2 | equequ1 1646 | . 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 ↔ 𝑥 = 𝑧)) | |
3 | 1, 2 | sbieh 1721 | 1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1290 [wsb 1693 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 df-sb 1694 |
This theorem is referenced by: equsb3 1874 |
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