Theorem zfrepclf 4569

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

Hypotheses

Ref	Expression
zfrepclf.1
zfrepclf.2
zfrepclf.3

Assertion

Ref	Expression
zfrepclf

Distinct variable groups:   ,,   ,   ,,

Proof of Theorem zfrepclf

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2
2		zfrepclf.1	. . . . . 6
3	2	nfeq2 2636	. . . . 5
4		eleq2 2530	. . . . . 6
5		zfrepclf.3	. . . . . 6
6	4, 5	syl6bi 228	. . . . 5
7	3, 6	alrimi 1877	. . . 4
8		nfv 1707	. . . . 5
9	8	axrep5 4568	. . . 4
10	7, 9	syl 16	. . 3
11	4	anbi1d 704	. . . . . . 7
12	3, 11	exbid 1886	. . . . . 6
13	12	bibi2d 318	. . . . 5
14	13	albidv 1713	. . . 4
15	14	exbidv 1714	. . 3
16	10, 15	mpbid 210	. 2
17	1, 16	vtocle 3183	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  F/_wnfc 2605  cvv 3109

This theorem is referenced by:  zfrep3cl  4570  zfrep4  4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111