Theorem cbvrabv 2748

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)

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

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvrabv

Step	Hyp	Ref	Expression
1		nfcv 2329	. 2 |- F/_ x A
2		nfcv 2329	. 2 |- F/_ y A
3		nfv 1538	. 2 |- F/ y ph
4		nfv 1538	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2747	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   {crab 2469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474

This theorem is referenced by:  pwnss  4171  acexmidlemv  5886  exmidac  7222  genipv  7522  ltexpri  7626  suplocsrlempr  7820  suplocsr  7822  zsupssdc  11969  sqne2sq  12191  eulerth  12247  odzval  12255  pcprecl  12303  pcprendvds  12304  pcpremul  12307  pceulem  12308