Theorem zfrep3 1476

Description: Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us ph is analogous to a "function" from x to y (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set z that corresponds to the "image" of ph restricted to some other set w. The hypothesis says z must not be free in ph.

Hypothesis

Ref	Expression
zfrep3.1	|- (ph -> A.zph)

Assertion

Ref	Expression
zfrep3	|- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem zfrep3

Step	Hyp	Ref	Expression
1		19.37v 961	. . . . 5 |- (E.z(x e. w -> A.y(ph -> y = z)) <-> (x e. w -> E.zA.y(ph -> y = z)))
2		impexp 276	. . . . . . . 8 |- (((x e. w /\ ph) -> y = z) <-> (x e. w -> (ph -> y = z)))
3	2	bial 695	. . . . . . 7 |- (A.y((x e. w /\ ph) -> y = z) <-> A.y(x e. w -> (ph -> y = z)))
4		19.21v 942	. . . . . . 7 |- (A.y(x e. w -> (ph -> y = z)) <-> (x e. w -> A.y(ph -> y = z)))
5	3, 4	bitr2 152	. . . . . 6 |- ((x e. w -> A.y(ph -> y = z)) <-> A.y((x e. w /\ ph) -> y = z))
6	5	biex 733	. . . . 5 |- (E.z(x e. w -> A.y(ph -> y = z)) <-> E.zA.y((x e. w /\ ph) -> y = z))
7	1, 6	bitr3 153	. . . 4 |- ((x e. w -> E.zA.y(ph -> y = z)) <-> E.zA.y((x e. w /\ ph) -> y = z))
8	7	bial 695	. . 3 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) <-> A.xE.zA.y((x e. w /\ ph) -> y = z))
9		ax-17 925	. . . . 5 |- (x e. w -> A.z x e. w)
10		zfrep3.1	. . . . 5 |- (ph -> A.zph)
11	9, 10	hban 704	. . . 4 |- ((x e. w /\ ph) -> A.z(x e. w /\ ph))
12	11	zfrep2 1475	. . 3 |- (A.xE.zA.y((x e. w /\ ph) -> y = z) -> E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))))
13	8, 12	sylbi 174	. 2 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))))
14		anabs5 375	. . . . . 6 |- ((x e. w /\ (x e. w /\ ph)) <-> (x e. w /\ ph))
15	14	biex 733	. . . . 5 |- (E.x(x e. w /\ (x e. w /\ ph)) <-> E.x(x e. w /\ ph))
16	15	bibi2i 460	. . . 4 |- ((y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> (y e. z <-> E.x(x e. w /\ ph)))
17	16	bial 695	. . 3 |- (A.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> A.y(y e. z <-> E.x(x e. w /\ ph)))
18	17	biex 733	. 2 |- (E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))
19	13, 18	sylib 173	1 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  zfrepclf 1477

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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