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Theorem stoweidlem4 27421
Description: Lemma for stoweid 27480: a class variable replaces a set variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypothesis
Ref Expression
stoweidlem4.1  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
Assertion
Ref Expression
stoweidlem4  |-  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A )
Distinct variable groups:    x, t, B    x, A    x, T    ph, x
Allowed substitution hints:    ph( t)    A( t)    T( t)

Proof of Theorem stoweidlem4
StepHypRef Expression
1 eleq1 2447 . . . . 5  |-  ( x  =  B  ->  (
x  e.  RR  <->  B  e.  RR ) )
21anbi2d 685 . . . 4  |-  ( x  =  B  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  B  e.  RR ) ) )
3 simpl 444 . . . . . 6  |-  ( ( x  =  B  /\  t  e.  T )  ->  x  =  B )
43mpteq2dva 4236 . . . . 5  |-  ( x  =  B  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  B ) )
54eleq1d 2453 . . . 4  |-  ( x  =  B  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  B )  e.  A ) )
62, 5imbi12d 312 . . 3  |-  ( x  =  B  ->  (
( ( ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A
)  <->  ( ( ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A ) ) )
7 stoweidlem4.1 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
86, 7vtoclg 2954 . 2  |-  ( B  e.  RR  ->  (
( ph  /\  B  e.  RR )  ->  (
t  e.  T  |->  B )  e.  A ) )
98anabsi7 793 1  |-  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4207   RRcr 8922
This theorem is referenced by:  stoweidlem18  27435  stoweidlem19  27436  stoweidlem22  27439  stoweidlem32  27449  stoweidlem36  27453  stoweidlem40  27457  stoweidlem41  27458  stoweidlem55  27472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-opab 4208  df-mpt 4209
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