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Theorem repizf 4002
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 4001. It is identical to zfrep6 4003 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 2003 . . 3  |-  ( E! y ph  ->  E. y ph )
21ralimi 2467 . 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 4001 . 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 1417   E.wex 1449   E!weu 1973   A.wral 2388   E.wrex 2389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-coll 4001
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-eu 1976  df-ral 2393
This theorem is referenced by:  zfrep6  4003  repizf2  4044
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