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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 1499 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | 2 | sbh 1756 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 Ⅎwnf 1440 [wsb 1742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 |
This theorem is referenced by: sbf2 1758 sbequ5 1762 sbequ6 1763 sbt 1764 sblim 1937 moimv 2072 moanim 2080 sbabel 2326 nfcdeq 2934 oprcl 3767 |
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