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Mirrors > Home > ILE Home > Th. List > repizf | GIF version |
Description: Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 3863. It is identical to zfrep6 3865 except for the choice of a freeness hypothesis rather than a distinct variable constraint between 𝑏 and φ. (Contributed by Jim Kingdon, 23-Aug-2018.) |
Ref | Expression |
---|---|
ax-coll.1 | ⊢ Ⅎ𝑏φ |
Ref | Expression |
---|---|
repizf | ⊢ (∀x ∈ 𝑎 ∃!yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 1927 | . . 3 ⊢ (∃!yφ → ∃yφ) | |
2 | 1 | ralimi 2378 | . 2 ⊢ (∀x ∈ 𝑎 ∃!yφ → ∀x ∈ 𝑎 ∃yφ) |
3 | ax-coll.1 | . . 3 ⊢ Ⅎ𝑏φ | |
4 | 3 | ax-coll 3863 | . 2 ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) |
5 | 2, 4 | syl 14 | 1 ⊢ (∀x ∈ 𝑎 ∃!yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1346 ∃wex 1378 ∃!weu 1897 ∀wral 2300 ∃wrex 2301 |
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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-coll 3863 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-eu 1900 df-ral 2305 |
This theorem is referenced by: zfrep6 3865 repizf2 3906 |
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