Theorem cbvrabv 1911

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvrabv.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvrabv	|- {x e. A | ph} = {y e. A | ps}

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

Proof of Theorem cbvrabv

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (z e. A -> A.x z e. A)
2		ax-17 971	. 2 |- (z e. A -> A.y z e. A)
3		ax-17 971	. 2 |- (ph -> A.yph)
4		ax-17 971	. 2 |- (ps -> A.xps)
5		cbvrabv.1	. 2 |- (x = y -> (ph <-> ps))
6	1, 2, 3, 4, 5	cbvrab 1910	1 |- {x e. A | ph} = {y e. A | ps}

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {crab 1648

This theorem is referenced by:  reuuni3 2886  inf3lema 4609  zorn2 4796  uzwo3lem2 6217  sqrlem24 6696  sqrgt0i 6697  sqrlem26 6698  seq1ub 6912  acdc3 7487  acdc2 7490  acdc5 7493  acdc 7495  pilem3 8673  pilem4 8674  nmcopex 9957  nmcfnex 9986  cnlnadj 10009  nmopadjle 10021

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-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

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-rab 1652  df-v 1812

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