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Theorem repizf 3847
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 3846. It is identical to zfrep6 3848 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 (x 𝑎 ∃!yφ𝑏x 𝑎 y 𝑏 φ)
Distinct variable group:   x,y,𝑎,𝑏
Allowed substitution hints:   φ(x,y,𝑎,𝑏)

Proof of Theorem repizf
StepHypRef Expression
1 euex 1912 . . 3 (∃!yφyφ)
21ralimi 2362 . 2 (x 𝑎 ∃!yφx 𝑎 yφ)
3 ax-coll.1 . . 3 𝑏φ
43ax-coll 3846 . 2 (x 𝑎 yφ𝑏x 𝑎 y 𝑏 φ)
52, 4syl 14 1 (x 𝑎 ∃!yφ𝑏x 𝑎 y 𝑏 φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1329  wex 1362  ∃!weu 1882  wral 2284  wrex 2285
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-coll 3846
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-ral 2289
This theorem is referenced by:  zfrep6  3848  repizf2  3889
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