Theorem zfrep3cl 4778

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
zfrep3cl.1	|- A e. _V
zfrep3cl.2	|- ( x e. A -> E. z A. y ( ph -> y = z ) )

Assertion

Ref	Expression
zfrep3cl	|- E. z A. y ( y e. z <-> E. x ( x e. A /\ ph ) )

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem zfrep3cl

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ x A
2		zfrep3cl.1	. 2 |- A e. _V
3		zfrep3cl.2	. 2 |- ( x e. A -> E. z A. y ( ph -> y = z ) )
4	1, 2, 3	zfrepclf 4777	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   E.wex 1704   e. wcel 1990   _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: (None)