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Theorem repizf 4137
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 4136. It is identical to zfrep6 4138 except for the choice of a freeness hypothesis rather than a disjoint variable condition 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 2068 . . 3  |-  ( E! y ph  ->  E. y ph )
21ralimi 2553 . 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 4136 . 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 1471   E.wex 1503   E!weu 2038   A.wral 2468   E.wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-coll 4136
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2041  df-ral 2473
This theorem is referenced by:  zfrep6  4138  repizf2  4183
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