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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 1412 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | 2 | sbh 1659 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 Ⅎwnf 1349 [wsb 1645 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: sbf2 1661 sbequ5 1665 sbequ6 1666 sbt 1667 sblim 1831 moimv 1966 moanim 1974 sbabel 2203 nfcdeq 2761 oprcl 3573 |
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