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Theorem repizf 3873
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 3872. It is identical to zfrep6 3874 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.)
Hypothesis
Ref Expression
ax-coll.1 𝑏𝜑
Assertion
Ref Expression
repizf (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
Distinct variable group:   𝑥,𝑦,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem repizf
StepHypRef Expression
1 euex 1930 . . 3 (∃!𝑦𝜑 → ∃𝑦𝜑)
21ralimi 2384 . 2 (∀𝑥𝑎 ∃!𝑦𝜑 → ∀𝑥𝑎𝑦𝜑)
3 ax-coll.1 . . 3 𝑏𝜑
43ax-coll 3872 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
52, 4syl 14 1 (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1349  wex 1381  ∃!weu 1900  wral 2306  wrex 2307
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-coll 3872
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-ral 2311
This theorem is referenced by:  zfrep6  3874  repizf2  3915
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