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Theorem repizf 4052
 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 4051. It is identical to zfrep6 4053 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 2030 . . 3 (∃!𝑦𝜑 → ∃𝑦𝜑)
21ralimi 2498 . 2 (∀𝑥𝑎 ∃!𝑦𝜑 → ∀𝑥𝑎𝑦𝜑)
3 ax-coll.1 . . 3 𝑏𝜑
43ax-coll 4051 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
52, 4syl 14 1 (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1437  ∃wex 1469  ∃!weu 2000  ∀wral 2417  ∃wrex 2418 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-coll 4051 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-ral 2422 This theorem is referenced by:  zfrep6  4053  repizf2  4094
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