![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > Mathboxes > setindf | GIF version |
Description: Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.) |
Ref | Expression |
---|---|
setindf.nf | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
setindf | ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setindft 15101 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)) | |
2 | setindf.nf | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | 1, 2 | mpg 1462 | 1 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 Ⅎwnf 1471 [wsb 1773 ∀wral 2468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-cleq 2182 df-clel 2185 df-ral 2473 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |