Theorem zfrepclf 4777

Description: An inference rule based on the Axiom of Replacement. Typically, ph defines a function from x to y. (Contributed by NM, 26-Nov-1995.)

Hypotheses

Ref	Expression
zfrepclf.1	|- F/_ x A
zfrepclf.2	|- A e. _V
zfrepclf.3	|- ( x e. A -> E. z A. y ( ph -> y = z ) )

Assertion

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

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

Allowed substitution hints:   ph( x, y)   A( x)

Proof of Theorem zfrepclf

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2 |- A e. _V
2		zfrepclf.1	. . . . . 6 |- F/_ x A
3	2	nfeq2 2780	. . . . 5 |- F/ x v = A
4		eleq2 2690	. . . . . 6 |- ( v = A -> ( x e. v <-> x e. A ) )
5		zfrepclf.3	. . . . . 6 |- ( x e. A -> E. z A. y ( ph -> y = z ) )
6	4, 5	syl6bi 243	. . . . 5 |- ( v = A -> ( x e. v -> E. z A. y ( ph -> y = z ) ) )
7	3, 6	alrimi 2082	. . . 4 |- ( v = A -> A. x ( x e. v -> E. z A. y ( ph -> y = z ) ) )
8		nfv 1843	. . . . 5 |- F/ z ph
9	8	axrep5 4776	. . . 4 |- ( A. x ( x e. v -> E. z A. y ( ph -> y = z ) ) -> E. z A. y ( y e. z <-> E. x ( x e. v /\ ph ) ) )
10	7, 9	syl 17	. . 3 |- ( v = A -> E. z A. y ( y e. z <-> E. x ( x e. v /\ ph ) ) )
11	4	anbi1d 741	. . . . . . 7 |- ( v = A -> ( ( x e. v /\ ph ) <-> ( x e. A /\ ph ) ) )
12	3, 11	exbid 2091	. . . . . 6 |- ( v = A -> ( E. x ( x e. v /\ ph ) <-> E. x ( x e. A /\ ph ) ) )
13	12	bibi2d 332	. . . . 5 |- ( v = A -> ( ( y e. z <-> E. x ( x e. v /\ ph ) ) <-> ( y e. z <-> E. x ( x e. A /\ ph ) ) ) )
14	13	albidv 1849	. . . 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 ) ) ) )
15	14	exbidv 1850	. . 3 |- ( v = A -> ( E. z A. y ( y e. z <-> E. x ( x e. v /\ ph ) ) <-> E. z A. y ( y e. z <-> E. x ( x e. A /\ ph ) ) ) )
16	10, 15	mpbid 222	. 2 |- ( v = A -> E. z A. y ( y e. z <-> E. x ( x e. A /\ ph ) ) )
17	1, 16	vtocle 3282	1 |- E. z A. y ( y e. z <-> E. x ( x e. A /\ ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  zfrep3cl  4778  zfrep4  4779