![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sblbis | GIF version |
Description: Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
Ref | Expression |
---|---|
sblbis.1 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sblbis | ⊢ ([𝑦 / 𝑥](𝜒 ↔ 𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbbi 1881 | . 2 ⊢ ([𝑦 / 𝑥](𝜒 ↔ 𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ [𝑦 / 𝑥]𝜑)) | |
2 | sblbis.1 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | |
3 | 2 | bibi2i 225 | . 2 ⊢ (([𝑦 / 𝑥]𝜒 ↔ [𝑦 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)) |
4 | 1, 3 | bitri 182 | 1 ⊢ ([𝑦 / 𝑥](𝜒 ↔ 𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 [wsb 1692 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 |
This theorem is referenced by: sb8eu 1961 sb8euh 1971 sb8iota 4987 |
Copyright terms: Public domain | W3C validator |