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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 4091. It is identical to zfrep6 4093 except for the choice of a freeness hypothesis rather than a disjoint variable condition 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 2043 | . . 3 ⊢ (∃!𝑦𝜑 → ∃𝑦𝜑) | |
2 | 1 | ralimi 2527 | . 2 ⊢ (∀𝑥 ∈ 𝑎 ∃!𝑦𝜑 → ∀𝑥 ∈ 𝑎 ∃𝑦𝜑) |
3 | ax-coll.1 | . . 3 ⊢ Ⅎ𝑏𝜑 | |
4 | 3 | ax-coll 4091 | . 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 1447 ∃wex 1479 ∃!weu 2013 ∀wral 2442 ∃wrex 2443 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-coll 4091 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-eu 2016 df-ral 2447 |
This theorem is referenced by: zfrep6 4093 repizf2 4135 |
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