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Mirrors > Home > ILE Home > Th. List > sbf | GIF version |
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sbf.1 | ⊢ Ⅎxφ |
Ref | Expression |
---|---|
sbf | ⊢ ([y / x]φ ↔ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbf.1 | . . 3 ⊢ Ⅎxφ | |
2 | 1 | nfri 1409 | . 2 ⊢ (φ → ∀xφ) |
3 | 2 | sbh 1656 | 1 ⊢ ([y / x]φ ↔ φ) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 Ⅎwnf 1346 [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-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: sbf2 1658 sbequ5 1662 sbequ6 1663 sbt 1664 sblim 1828 moimv 1963 moanim 1971 sbabel 2200 nfcdeq 2755 oprcl 3564 |
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