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Theorem repizf 4105
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 4104. It is identical to zfrep6 4106 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.)
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 2049 . . 3 (∃!𝑦𝜑 → ∃𝑦𝜑)
21ralimi 2533 . 2 (∀𝑥𝑎 ∃!𝑦𝜑 → ∀𝑥𝑎𝑦𝜑)
3 ax-coll.1 . . 3 𝑏𝜑
43ax-coll 4104 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
52, 4syl 14 1 (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1453  wex 1485  ∃!weu 2019  wral 2448  wrex 2449
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-coll 4104
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-ral 2453
This theorem is referenced by:  zfrep6  4106  repizf2  4148
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