Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > equsb3ALT | Structured version Visualization version GIF version |
Description: Alternate proof of equsb3 2420, shorter but requiring ax-11 2021. (Contributed by Raph Levien and FL, 4-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
equsb3ALT | ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb3lem 2419 | . . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧) | |
2 | 1 | sbbii 1874 | . 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧) |
3 | nfv 1830 | . . 3 ⊢ Ⅎ𝑤 𝑦 = 𝑧 | |
4 | 3 | sbco2 2403 | . 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧) |
5 | equsb3lem 2419 | . 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧) | |
6 | 2, 4, 5 | 3bitr3i 289 | 1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |