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Mirrors > Home > ILE Home > Th. List > sb8 | GIF version |
Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Ref | Expression |
---|---|
sb8e.1 | ⊢ Ⅎyφ |
Ref | Expression |
---|---|
sb8 | ⊢ (∀xφ ↔ ∀y[y / x]φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8e.1 | . 2 ⊢ Ⅎyφ | |
2 | 1 | nfs1 1687 | . 2 ⊢ Ⅎx[y / x]φ |
3 | sbequ12 1651 | . 2 ⊢ (x = y → (φ ↔ [y / x]φ)) | |
4 | 1, 2, 3 | cbval 1634 | 1 ⊢ (∀xφ ↔ ∀y[y / x]φ) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1240 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-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 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: sbnf2 1854 sb8eu 1910 nfraldya 2352 rabeq0 3241 abeq0 3242 sb8iota 4817 |
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