Theorem cbvrabv 2802

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 2375	. 2 |- F/_ x A
2		nfcv 2375	. 2 |- F/_ y A
3		nfv 1577	. 2 |- F/ y ph
4		nfv 1577	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2801	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   {crab 2515

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520

This theorem is referenced by:  pwnss  4255  acexmidlemv  6026  exmidac  7484  genipv  7789  ltexpri  7893  suplocsrlempr  8087  suplocsr  8089  zsupssdc  10561  bitsfzolem  12595  nninfctlemfo  12691  sqne2sq  12829  eulerth  12885  odzval  12894  pcprecl  12942  pcprendvds  12943  pcpremul  12946  pceulem  12947  4sqlem19  13062  lfgredg2dom  16073  vtxdumgrfival  16239  vtxduspgrfvedgfilem  16241  vtxduspgrfvedgfi  16242