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Theorem raleqfnOLD 26357
Description: Change the domain of quantification by a function. (Moved to ralima 5758 in main set.mm and may be deleted by mathbox owner, JM. --NM 17-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
raleqfnOLD.1  |-  ( y  =  ( F `  x )  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
raleqfnOLD  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( A. x  e.  B  ps  <->  A. y  e.  ( F " B
) ph ) )
Distinct variable groups:    x, F, y    x, A, y    x, B, y    ph, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)

Proof of Theorem raleqfnOLD
StepHypRef Expression
1 raleqfnOLD.1 . . 3  |-  ( y  =  ( F `  x )  ->  ( ph 
<->  ps ) )
21ralima 5758 . 2  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( A. y  e.  ( F " B
) ph  <->  A. x  e.  B  ps ) )
32bicomd 192 1  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( A. x  e.  B  ps  <->  A. y  e.  ( F " B
) ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543    C_ wss 3152   "cima 4692    Fn wfn 5250   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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