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Theorem stoweidlem4 37804
Description: Lemma for stoweid 37865: 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 2495 . . . . 5  |-  ( x  =  B  ->  (
x  e.  RR  <->  B  e.  RR ) )
21anbi2d 708 . . . 4  |-  ( x  =  B  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  B  e.  RR ) ) )
3 simpl 458 . . . . . 6  |-  ( ( x  =  B  /\  t  e.  T )  ->  x  =  B )
43mpteq2dva 4510 . . . . 5  |-  ( x  =  B  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  B ) )
54eleq1d 2491 . . . 4  |-  ( x  =  B  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  B )  e.  A ) )
62, 5imbi12d 321 . . 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 3139 . 2  |-  ( B  e.  RR  ->  (
( ph  /\  B  e.  RR )  ->  (
t  e.  T  |->  B )  e.  A ) )
98anabsi7 826 1  |-  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    |-> cmpt 4482   RRcr 9545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-ral 2776  df-v 3082  df-opab 4483  df-mpt 4484
This theorem is referenced by:  stoweidlem18  37818  stoweidlem19  37819  stoweidlem22  37822  stoweidlem32  37833  stoweidlem36  37837  stoweidlem40  37841  stoweidlem41  37842  stoweidlem55  37856
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