Theorem cbvabv 2473

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)

Hypothesis

Ref	Expression
cbvabv.1	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvabv	|- {x | ph} = {y | ps}

Distinct variable groups:   ph,y   ps,x

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

Proof of Theorem cbvabv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/yph
2		nfv 1619	. 2 |- F/xps
3		cbvabv.1	. 2 |- (x = y -> (ph <-> ps))
4	1, 2, 3	cbvab 2472	1 |- {x | ph} = {y | ps}

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by:  ninjust  3211  uniiunlem  3354  dfif3  3673  pwjust  3724  snjust  3741  intab  3957  iotajust  4339