Theorem zfrep3cl 2697

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

Hypotheses

Ref	Expression
zfrep3cl.1	|- A e. V
zfrep3cl.2	|- (x e. A -> E.zA.y(ph -> y = z))

Assertion

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

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

Proof of Theorem zfrep3cl

Step	Hyp	Ref	Expression
1		ax-17 970	. 2 |- (y e. A -> A.x y e. A)
2		zfrep3cl.1	. 2 |- A e. V
3		zfrep3cl.2	. 2 |- (x e. A -> E.zA.y(ph -> y = z))
4	1, 2, 3	zfrepclf 2696	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 1809

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 1459  ax-rep 2690

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810

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