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Theorem ralrabOLD 26458
Description: Universal quantification over a restricted class abstraction. (Moved to ralrab 2940 in main set.mm and may be deleted by mathbox owner, JM. --NM 20-Oct-2011.) (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ralab.1OLD  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralrabOLD  |-  ( A. x  e.  { y  e.  A  |  ph } ch 
<-> 
A. x  e.  A  ( ps  ->  ch )
)
Distinct variable groups:    x, y    ps, y    x, A, y    ph, x
Allowed substitution hints:    ph( y)    ps( x)    ch( x, y)

Proof of Theorem ralrabOLD
StepHypRef Expression
1 ralab.1OLD . 2  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
21ralrab 2940 1  |-  ( A. x  e.  { y  e.  A  |  ph } ch 
<-> 
A. x  e.  A  ( ps  ->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wral 2556   {crab 2560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803
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