Theorem cbvrabv 2801

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 2374	. 2 |- F/_ x A
2		nfcv 2374	. 2 |- F/_ y A
3		nfv 1576	. 2 |- F/ y ph
4		nfv 1576	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2800	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   {crab 2514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519

This theorem is referenced by:  pwnss  4249  acexmidlemv  6016  exmidac  7424  genipv  7729  ltexpri  7833  suplocsrlempr  8027  suplocsr  8029  zsupssdc  10499  bitsfzolem  12533  nninfctlemfo  12629  sqne2sq  12767  eulerth  12823  odzval  12832  pcprecl  12880  pcprendvds  12881  pcpremul  12884  pceulem  12885  4sqlem19  13000  lfgredg2dom  16002  vtxdumgrfival  16168  vtxduspgrfvedgfilem  16170  vtxduspgrfvedgfi  16171