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Theorem frege116 36646
Description: One direction of dffrege115 36645. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege116.x  |-  X  e.  U
Assertion
Ref Expression
frege116  |-  ( Fun  `' `' R  ->  A. b
( b R X  ->  A. a ( b R a  ->  a  =  X ) ) )
Distinct variable groups:    a, b, R    X, a, b
Allowed substitution hints:    U( a, b)

Proof of Theorem frege116
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 dffrege115 36645 . 2  |-  ( A. c A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  Fun  `' `' R )
2 frege116.x . . . 4  |-  X  e.  U
32frege68c 36598 . . 3  |-  ( ( A. c A. b
( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  Fun  `' `' R )  ->  ( Fun  `' `' R  ->  [. X  /  c ]. A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) ) ) )
4 sbcal 3305 . . . 4  |-  ( [. X  /  c ]. A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  A. b [. X  /  c ]. ( b R c  ->  A. a ( b R a  ->  a  =  c ) ) )
5 sbcimg 3297 . . . . . . 7  |-  ( X  e.  U  ->  ( [. X  /  c ]. ( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <-> 
( [. X  /  c ]. b R c  ->  [. X  /  c ]. A. a ( b R a  ->  a  =  c ) ) ) )
62, 5ax-mp 5 . . . . . 6  |-  ( [. X  /  c ]. (
b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  ( [. X  /  c ]. b R c  ->  [. X  /  c ]. A. a ( b R a  ->  a  =  c ) ) )
7 sbcbr2g 4451 . . . . . . . . 9  |-  ( X  e.  U  ->  ( [. X  /  c ]. b R c  <->  b R [_ X  /  c ]_ c ) )
82, 7ax-mp 5 . . . . . . . 8  |-  ( [. X  /  c ]. b R c  <->  b R [_ X  /  c ]_ c )
9 csbvarg 3796 . . . . . . . . . 10  |-  ( X  e.  U  ->  [_ X  /  c ]_ c  =  X )
102, 9ax-mp 5 . . . . . . . . 9  |-  [_ X  /  c ]_ c  =  X
1110breq2i 4403 . . . . . . . 8  |-  ( b R [_ X  / 
c ]_ c  <->  b R X )
128, 11bitri 257 . . . . . . 7  |-  ( [. X  /  c ]. b R c  <->  b R X )
13 sbcal 3305 . . . . . . . 8  |-  ( [. X  /  c ]. A. a ( b R a  ->  a  =  c )  <->  A. a [. X  /  c ]. ( b R a  ->  a  =  c ) )
14 sbcimg 3297 . . . . . . . . . . 11  |-  ( X  e.  U  ->  ( [. X  /  c ]. ( b R a  ->  a  =  c )  <->  ( [. X  /  c ]. b R a  ->  [. X  /  c ]. a  =  c ) ) )
152, 14ax-mp 5 . . . . . . . . . 10  |-  ( [. X  /  c ]. (
b R a  -> 
a  =  c )  <-> 
( [. X  /  c ]. b R a  ->  [. X  /  c ]. a  =  c
) )
16 sbcg 3321 . . . . . . . . . . . 12  |-  ( X  e.  U  ->  ( [. X  /  c ]. b R a  <->  b R
a ) )
172, 16ax-mp 5 . . . . . . . . . . 11  |-  ( [. X  /  c ]. b R a  <->  b R
a )
18 sbceq2g 3783 . . . . . . . . . . . . 13  |-  ( X  e.  U  ->  ( [. X  /  c ]. a  =  c  <->  a  =  [_ X  / 
c ]_ c ) )
192, 18ax-mp 5 . . . . . . . . . . . 12  |-  ( [. X  /  c ]. a  =  c  <->  a  =  [_ X  /  c ]_ c
)
2010eqeq2i 2483 . . . . . . . . . . . 12  |-  ( a  =  [_ X  / 
c ]_ c  <->  a  =  X )
2119, 20bitri 257 . . . . . . . . . . 11  |-  ( [. X  /  c ]. a  =  c  <->  a  =  X )
2217, 21imbi12i 333 . . . . . . . . . 10  |-  ( (
[. X  /  c ]. b R a  ->  [. X  /  c ]. a  =  c
)  <->  ( b R a  ->  a  =  X ) )
2315, 22bitri 257 . . . . . . . . 9  |-  ( [. X  /  c ]. (
b R a  -> 
a  =  c )  <-> 
( b R a  ->  a  =  X ) )
2423albii 1699 . . . . . . . 8  |-  ( A. a [. X  /  c ]. ( b R a  ->  a  =  c )  <->  A. a ( b R a  ->  a  =  X ) )
2513, 24bitri 257 . . . . . . 7  |-  ( [. X  /  c ]. A. a ( b R a  ->  a  =  c )  <->  A. a
( b R a  ->  a  =  X ) )
2612, 25imbi12i 333 . . . . . 6  |-  ( (
[. X  /  c ]. b R c  ->  [. X  /  c ]. A. a ( b R a  ->  a  =  c ) )  <-> 
( b R X  ->  A. a ( b R a  ->  a  =  X ) ) )
276, 26bitri 257 . . . . 5  |-  ( [. X  /  c ]. (
b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  ( b R X  ->  A. a
( b R a  ->  a  =  X ) ) )
2827albii 1699 . . . 4  |-  ( A. b [. X  /  c ]. ( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  A. b ( b R X  ->  A. a
( b R a  ->  a  =  X ) ) )
294, 28bitri 257 . . 3  |-  ( [. X  /  c ]. A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  A. b
( b R X  ->  A. a ( b R a  ->  a  =  X ) ) )
303, 29syl6ib 234 . 2  |-  ( ( A. c A. b
( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  Fun  `' `' R )  ->  ( Fun  `' `' R  ->  A. b
( b R X  ->  A. a ( b R a  ->  a  =  X ) ) ) )
311, 30ax-mp 5 1  |-  ( Fun  `' `' R  ->  A. b
( b R X  ->  A. a ( b R a  ->  a  =  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452    e. wcel 1904   [.wsbc 3255   [_csb 3349   class class class wbr 4395   `'ccnv 4838   Fun wfun 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-frege1 36457  ax-frege2 36458  ax-frege8 36476  ax-frege52a 36524  ax-frege58b 36568
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5591
This theorem is referenced by:  frege117  36647
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