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Mirrors > Home > ILE Home > Th. List > equsb3lem | GIF version |
Description: Lemma for equsb3 1951. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
equsb3lem | ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1526 | . 2 ⊢ (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧) | |
2 | equequ1 1712 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
3 | 1, 2 | sbieh 1790 | 1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 [wsb 1762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-sb 1763 |
This theorem is referenced by: equsb3 1951 |
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