Theorem cbvrabv 2759

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 2336	. 2 |- F/_ x A
2		nfcv 2336	. 2 |- F/_ y A
3		nfv 1539	. 2 |- F/ y ph
4		nfv 1539	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2758	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   {crab 2476

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481

This theorem is referenced by:  pwnss  4189  acexmidlemv  5917  exmidac  7271  genipv  7571  ltexpri  7675  suplocsrlempr  7869  suplocsr  7871  zsupssdc  12094  nninfctlemfo  12180  sqne2sq  12318  eulerth  12374  odzval  12382  pcprecl  12430  pcprendvds  12431  pcpremul  12434  pceulem  12435  4sqlem19  12550