Theorem repizf 4176

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 4175. It is identical to zfrep6 4177 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

Step	Hyp	Ref	Expression
1		euex 2085	. . 3 |- ( E! y ph -> E. y ph )
2	1	ralimi 2571	. 2 |- ( A. x e. a E! y ph -> A. x e. a E. y ph )
3		ax-coll.1	. . 3 |- F/ b ph
4	3	ax-coll 4175	. 2 |- ( A. x e. a E. y ph -> E. b A. x e. a E. y e. b ph )
5	2, 4	syl 14	1 |- ( A. x e. a E! y ph -> E. b A. x e. a E. y e. b ph )

Syntax hints:   -> wi 4   F/wnf 1484   E.wex 1516   E!weu 2055   A.wral 2486   E.wrex 2487

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-coll 4175

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-ral 2491

This theorem is referenced by:  zfrep6  4177  repizf2  4222