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Theorem repizf 3914
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 3913. It is identical to zfrep6 3915 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 1973 . . 3 (∃!𝑦𝜑 → ∃𝑦𝜑)
21ralimi 2431 . 2 (∀𝑥𝑎 ∃!𝑦𝜑 → ∀𝑥𝑎𝑦𝜑)
3 ax-coll.1 . . 3 𝑏𝜑
43ax-coll 3913 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
52, 4syl 14 1 (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1390  wex 1422  ∃!weu 1943  wral 2353  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-coll 3913
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-ral 2358
This theorem is referenced by:  zfrep6  3915  repizf2  3956
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