Theorem zfrepclf 2699

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-17 971	. . . . . 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 1565	. . . . 5 |- (v = A -> A.x v = A)
5		eleq2 1535	. . . . . 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 998	. . . 4 |- (v = A -> A.x(x e. v -> E.zA.y(ph -> y = z)))
9		ax-17 971	. . . . 5 |- (ph -> A.zph)
10	9	axrep5 2698	. . . 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 617	. . . . . . 7 |- (v = A -> ((x e. v /\ ph) <-> (x e. A /\ ph)))
13	4, 12	exbid 1105	. . . . . 6 |- (v = A -> (E.x(x e. v /\ ph) <-> E.x(x e. A /\ ph)))
14	13	bibi2d 618	. . . . 5 |- (v = A -> ((y e. z <-> E.x(x e. v /\ ph)) <-> (y e. z <-> E.x(x e. A /\ ph))))
15	14	albidv 1278	. . . 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 1279	. . 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 1858	1 |- E.zA.y(y e. z <-> E.x(x e. A /\ ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  zfrep3cl 2700  zfrep4 2701

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459  ax-rep 2693

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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