Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqsb3 | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff (class version of equsb3 2420). (Contributed by Rodolfo Medina, 28-Apr-2010.) |
Ref | Expression |
---|---|
eqsb3 | ⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsb3lem 2714 | . . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝐴 ↔ 𝑤 = 𝐴) | |
2 | 1 | sbbii 1874 | . 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴) |
3 | nfv 1830 | . . 3 ⊢ Ⅎ𝑤 𝑦 = 𝐴 | |
4 | 3 | sbco2 2403 | . 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴) |
5 | eqsb3lem 2714 | . 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝐴 ↔ 𝑥 = 𝐴) | |
6 | 2, 4, 5 | 3bitr3i 289 | 1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 [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 ax-ext 2590 |
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 df-cleq 2603 |
This theorem is referenced by: pm13.183 3313 eqsbc3 3442 |
Copyright terms: Public domain | W3C validator |