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Mirrors > Home > MPE Home > Th. List > equsb3lem | Structured version Visualization version GIF version |
Description: Lemma for equsb3 2565. (Contributed by Raph Levien and FL, 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 | nfv 1988 | . 2 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
2 | equequ1 2103 | . 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 ↔ 𝑥 = 𝑧)) | |
3 | 1, 2 | sbie 2541 | 1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 [wsb 2042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-10 2164 ax-12 2192 ax-13 2387 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ex 1850 df-nf 1855 df-sb 2043 |
This theorem is referenced by: equsb3 2565 equsb3ALT 2566 |
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