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Theorem stoweidlem4 31952
Description: Lemma for stoweid 32011: a class variable replaces a setvar 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 2454 . . . . 5  |-  ( x  =  B  ->  (
x  e.  RR  <->  B  e.  RR ) )
21anbi2d 701 . . . 4  |-  ( x  =  B  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  B  e.  RR ) ) )
3 simpl 455 . . . . . 6  |-  ( ( x  =  B  /\  t  e.  T )  ->  x  =  B )
43mpteq2dva 4453 . . . . 5  |-  ( x  =  B  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  B ) )
54eleq1d 2451 . . . 4  |-  ( x  =  B  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  B )  e.  A ) )
62, 5imbi12d 318 . . 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 3092 . 2  |-  ( B  e.  RR  ->  (
( ph  /\  B  e.  RR )  ->  (
t  e.  T  |->  B )  e.  A ) )
98anabsi7 817 1  |-  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    |-> cmpt 4425   RRcr 9402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-ral 2737  df-v 3036  df-opab 4426  df-mpt 4427
This theorem is referenced by:  stoweidlem18  31966  stoweidlem19  31967  stoweidlem22  31970  stoweidlem32  31980  stoweidlem36  31984  stoweidlem40  31988  stoweidlem41  31989  stoweidlem55  32003
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