![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbrim | GIF version |
Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbrim.1 | ⊢ (φ → ∀xφ) |
Ref | Expression |
---|---|
sbrim | ⊢ ([y / x](φ → ψ) ↔ (φ → [y / x]ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 1824 | . 2 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ)) | |
2 | sbrim.1 | . . . 4 ⊢ (φ → ∀xφ) | |
3 | 2 | sbh 1656 | . . 3 ⊢ ([y / x]φ ↔ φ) |
4 | 3 | imbi1i 227 | . 2 ⊢ (([y / x]φ → [y / x]ψ) ↔ (φ → [y / x]ψ)) |
5 | 1, 4 | bitri 173 | 1 ⊢ ([y / x](φ → ψ) ↔ (φ → [y / x]ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: sbco2d 1837 sbco2vd 1838 hbsbd 1855 |
Copyright terms: Public domain | W3C validator |