Theorem cbvrabv 2738

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 2319	. 2 |- F/_ x A
2		nfcv 2319	. 2 |- F/_ y A
3		nfv 1528	. 2 |- F/ y ph
4		nfv 1528	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2737	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   {crab 2459

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464

This theorem is referenced by:  pwnss  4161  acexmidlemv  5876  exmidac  7211  genipv  7511  ltexpri  7615  suplocsrlempr  7809  suplocsr  7811  zsupssdc  11958  sqne2sq  12180  eulerth  12236  odzval  12244  pcprecl  12292  pcprendvds  12293  pcpremul  12296  pceulem  12297