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

Step	Hyp	Ref	Expression
1		euex 1972	. . 3 |- ( E! y ph -> E. y ph )
2	1	ralimi 2427	. 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 3901	. 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 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