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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 1802 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 1764 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbval 1747 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1346 Ⅎwnf 1453 [wsb 1755 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 |
This theorem is referenced by: sbnf2 1974 sb8eu 2032 nfraldya 2505 rabeq0 3443 abeq0 3444 sb8iota 5165 |
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