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Mirrors > Home > ILE Home > Th. List > Mathboxes > setindf | GIF version |
Description: Axiom of set-induction with a disjoint variable condition replaced with a non-freeness 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 13152 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)) | |
2 | setindf.nf | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | 1, 2 | mpg 1427 | 1 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 Ⅎwnf 1436 [wsb 1735 ∀wral 2414 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-cleq 2130 df-clel 2133 df-ral 2419 |
This theorem is referenced by: (None) |
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