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Theorem elimhyps 32446
Description: A version of elimhyp 3964 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
Hypothesis
Ref Expression
elimhyps.1  |-  [. B  /  x ]. ph
Assertion
Ref Expression
elimhyps  |-  [. if ( ph ,  x ,  B )  /  x ]. ph

Proof of Theorem elimhyps
StepHypRef Expression
1 dfsbcq 3298 . . 3  |-  ( x  =  if ( [. x  /  x ]. ph ,  x ,  B )  ->  ( [. x  /  x ]. ph  <->  [. if (
[. x  /  x ]. ph ,  x ,  B )  /  x ]. ph ) )
2 dfsbcq 3298 . . 3  |-  ( B  =  if ( [. x  /  x ]. ph ,  x ,  B )  ->  ( [. B  /  x ]. ph  <->  [. if (
[. x  /  x ]. ph ,  x ,  B )  /  x ]. ph ) )
3 elimhyps.1 . . 3  |-  [. B  /  x ]. ph
41, 2, 3elimhyp 3964 . 2  |-  [. if ( [. x  /  x ]. ph ,  x ,  B )  /  x ]. ph
5 sbcid 3313 . . 3  |-  ( [. x  /  x ]. ph  <->  ph )
6 ifbi 3927 . . 3  |-  ( (
[. x  /  x ]. ph  <->  ph )  ->  if ( [. x  /  x ]. ph ,  x ,  B )  =  if ( ph ,  x ,  B ) )
7 dfsbcq 3298 . . . 4  |-  ( if ( [. x  /  x ]. ph ,  x ,  B )  =  if ( ph ,  x ,  B )  ->  ( [. if ( [. x  /  x ]. ph ,  x ,  B )  /  x ]. ph  <->  [. if (
ph ,  x ,  B )  /  x ]. ph ) )
87bicomd 204 . . 3  |-  ( if ( [. x  /  x ]. ph ,  x ,  B )  =  if ( ph ,  x ,  B )  ->  ( [. if ( ph ,  x ,  B )  /  x ]. ph  <->  [. if (
[. x  /  x ]. ph ,  x ,  B )  /  x ]. ph ) )
95, 6, 8mp2b 10 . 2  |-  ( [. if ( ph ,  x ,  B )  /  x ]. ph  <->  [. if ( [. x  /  x ]. ph ,  x ,  B )  /  x ]. ph )
104, 9mpbir 212 1  |-  [. if ( ph ,  x ,  B )  /  x ]. ph
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437   [.wsbc 3296   ifcif 3906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-sbc 3297  df-if 3907
This theorem is referenced by:  renegclALT  32448
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