Theorem repizf 4149

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 4148. It is identical to zfrep6 4150 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 2075	. . 3 |- ( E! y ph -> E. y ph )
2	1	ralimi 2560	. 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 4148	. 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 1474   E.wex 1506   E!weu 2045   A.wral 2475   E.wrex 2476

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-coll 4148

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-ral 2480

This theorem is referenced by:  zfrep6  4150  repizf2  4195