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Theorem stoweidlem4 27856
Description: Lemma for stoweid 27915: 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 simpr 447 . . 3  |-  ( (
ph  /\  B  e.  RR )  ->  B  e.  RR )
2 eleq1 2356 . . . . . 6  |-  ( x  =  B  ->  (
x  e.  RR  <->  B  e.  RR ) )
32anbi2d 684 . . . . 5  |-  ( x  =  B  ->  (
( ph  /\  x  e.  RR )  <->  ( ph  /\  B  e.  RR ) ) )
4 nfv 1609 . . . . . . 7  |-  F/ t  x  =  B
5 simpl 443 . . . . . . 7  |-  ( ( x  =  B  /\  t  e.  T )  ->  x  =  B )
64, 5mpteq2da 4121 . . . . . 6  |-  ( x  =  B  ->  (
t  e.  T  |->  x )  =  ( t  e.  T  |->  B ) )
76eleq1d 2362 . . . . 5  |-  ( x  =  B  ->  (
( t  e.  T  |->  x )  e.  A  <->  ( t  e.  T  |->  B )  e.  A ) )
83, 7imbi12d 311 . . . 4  |-  ( x  =  B  ->  (
( ( ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A
)  <->  ( ( ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A ) ) )
9 stoweidlem4.1 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
10 ax-1 5 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR )  ->  (
t  e.  T  |->  x )  e.  A )  ->  ( x  e.  RR  ->  ( ( ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A ) ) )
119, 10ax-mp 8 . . . 4  |-  ( x  e.  RR  ->  (
( ph  /\  x  e.  RR )  ->  (
t  e.  T  |->  x )  e.  A ) )
128, 11vtoclga 2862 . . 3  |-  ( B  e.  RR  ->  (
( ph  /\  B  e.  RR )  ->  (
t  e.  T  |->  B )  e.  A ) )
131, 12syl 15 . 2  |-  ( (
ph  /\  B  e.  RR )  ->  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A ) )
1413pm2.43i 43 1  |-  ( (
ph  /\  B  e.  RR )  ->  ( t  e.  T  |->  B )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   RRcr 8752
This theorem is referenced by:  stoweidlem18  27870  stoweidlem19  27871  stoweidlem22  27874  stoweidlem32  27884  stoweidlem36  27888  stoweidlem40  27892  stoweidlem41  27893  stoweidlem55  27907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-opab 4094  df-mpt 4095
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