Theorem zfrepclf 2695

Description: An inference rule based on the Axiom of Replacement. Typically, ph defines a function from x to y.

Hypotheses

Ref	Expression
zfrepclf.1	|- (w e. A -> A.x w e. A)
zfrepclf.2	|- A e. V
zfrepclf.3	|- (x e. A -> E.zA.y(ph -> y = z))

Assertion

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

Distinct variable groups:   y,z,A   ph,z   w,A   x,y,z   x,w

Proof of Theorem zfrepclf

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2 |- A e. V
2		ax-17 970	. . . . . 6 |- (w e. v -> A.x w e. v)
3		zfrepclf.1	. . . . . 6 |- (w e. A -> A.x w e. A)
4	2, 3	hbeq 1563	. . . . 5 |- (v = A -> A.x v = A)
5		eleq2 1533	. . . . . 6 |- (v = A -> (x e. v <-> x e. A))
6		zfrepclf.3	. . . . . 6 |- (x e. A -> E.zA.y(ph -> y = z))
7	5, 6	syl6bi 214	. . . . 5 |- (v = A -> (x e. v -> E.zA.y(ph -> y = z)))
8	4, 7	19.21ai 997	. . . 4 |- (v = A -> A.x(x e. v -> E.zA.y(ph -> y = z)))
9		ax-17 970	. . . . 5 |- (ph -> A.zph)
10	9	axrep5 2694	. . . 4 |- (A.x(x e. v -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. v /\ ph)))
11	8, 10	syl 10	. . 3 |- (v = A -> E.zA.y(y e. z <-> E.x(x e. v /\ ph)))
12	5	anbi1d 616	. . . . . . 7 |- (v = A -> ((x e. v /\ ph) <-> (x e. A /\ ph)))
13	4, 12	exbid 1104	. . . . . 6 |- (v = A -> (E.x(x e. v /\ ph) <-> E.x(x e. A /\ ph)))
14	13	bibi2d 617	. . . . 5 |- (v = A -> ((y e. z <-> E.x(x e. v /\ ph)) <-> (y e. z <-> E.x(x e. A /\ ph))))
15	14	albidv 1277	. . . 4 |- (v = A -> (A.y(y e. z <-> E.x(x e. v /\ ph)) <-> A.y(y e. z <-> E.x(x e. A /\ ph))))
16	15	exbidv 1278	. . 3 |- (v = A -> (E.zA.y(y e. z <-> E.x(x e. v /\ ph)) <-> E.zA.y(y e. z <-> E.x(x e. A /\ ph))))
17	11, 16	mpbid 195	. 2 |- (v = A -> E.zA.y(y e. z <-> E.x(x e. A /\ ph)))
18	1, 17	vtocle 1855	1 |- E.zA.y(y e. z <-> E.x(x e. A /\ ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808

This theorem is referenced by:  zfrep3cl 2696  zfrep4 2697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-12 967  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458  ax-rep 2689

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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