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Theorem stoweidlem4 29825
Description: Lemma for stoweid 29884: 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 2503 . . . . 5  |-  ( x  =  B  ->  (
x  e.  RR  <->  B  e.  RR ) )
21anbi2d 703 . . . 4  |-  ( x  =  B  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  B  e.  RR ) ) )
3 simpl 457 . . . . . 6  |-  ( ( x  =  B  /\  t  e.  T )  ->  x  =  B )
43mpteq2dva 4399 . . . . 5  |-  ( x  =  B  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  B ) )
54eleq1d 2509 . . . 4  |-  ( x  =  B  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  B )  e.  A ) )
62, 5imbi12d 320 . . 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 3051 . 2  |-  ( B  e.  RR  ->  (
( ph  /\  B  e.  RR )  ->  (
t  e.  T  |->  B )  e.  A ) )
98anabsi7 815 1  |-  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4371   RRcr 9302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-ral 2741  df-v 2995  df-opab 4372  df-mpt 4373
This theorem is referenced by:  stoweidlem18  29839  stoweidlem19  29840  stoweidlem22  29843  stoweidlem32  29853  stoweidlem36  29857  stoweidlem40  29861  stoweidlem41  29862  stoweidlem55  29876
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