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Theorem repizf 3902
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 3901. It is identical to zfrep6 3903 except for the choice of a freeness hypothesis rather than a distinct variable constraint between  b and  ph. (Contributed by Jim Kingdon, 23-Aug-2018.)
Hypothesis
Ref Expression
ax-coll.1  |-  F/ b
ph
Assertion
Ref Expression
repizf  |-  ( A. x  e.  a  E! y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
Distinct variable group:    x, y, a, b
Allowed substitution hints:    ph( x, y, a, b)

Proof of Theorem repizf
StepHypRef Expression
1 euex 1972 . . 3  |-  ( E! y ph  ->  E. y ph )
21ralimi 2427 . 2  |-  ( A. x  e.  a  E! y ph  ->  A. x  e.  a  E. y ph )
3 ax-coll.1 . . 3  |-  F/ b
ph
43ax-coll 3901 . 2  |-  ( A. x  e.  a  E. y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
52, 4syl 14 1  |-  ( A. x  e.  a  E! y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1390   E.wex 1422   E!weu 1942   A.wral 2349   E.wrex 2350
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 3901
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-ral 2354
This theorem is referenced by:  zfrep6  3903  repizf2  3944
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