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Theorem frege123 36626
Description: Lemma for frege124 36627. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege123.x  |-  X  e.  U
frege123.y  |-  Y  e.  V
Assertion
Ref Expression
frege123  |-  ( ( A. a ( Y R a  ->  X
( ( t+ `  R )  u.  _I  ) a )  ->  ( Y ( t+ `  R
) M  ->  X
( ( t+ `  R )  u.  _I  ) M ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `  R ) M  ->  X ( ( t+ `  R
)  u.  _I  ) M ) ) ) )
Distinct variable groups:    R, a    X, a    Y, a
Allowed substitution hints:    U( a)    M( a)    V( a)

Proof of Theorem frege123
StepHypRef Expression
1 frege123.x . . . 4  |-  X  e.  U
2 frege123.y . . . 4  |-  Y  e.  V
3 vex 3059 . . . 4  |-  a  e. 
_V
41, 2, 3frege122 36625 . . 3  |-  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y R a  ->  X
( ( t+ `  R )  u.  _I  ) a ) ) )
54alrimdv 1785 . 2  |-  ( Fun  `' `' R  ->  ( Y R X  ->  A. a
( Y R a  ->  X ( ( t+ `  R
)  u.  _I  )
a ) ) )
6 frege19 36464 . 2  |-  ( ( Fun  `' `' R  ->  ( Y R X  ->  A. a ( Y R a  ->  X
( ( t+ `  R )  u.  _I  ) a ) ) )  ->  (
( A. a ( Y R a  ->  X ( ( t+ `  R )  u.  _I  ) a )  ->  ( Y
( t+ `  R ) M  ->  X ( ( t+ `  R )  u.  _I  ) M ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `  R ) M  ->  X ( ( t+ `  R
)  u.  _I  ) M ) ) ) ) )
75, 6ax-mp 5 1  |-  ( ( A. a ( Y R a  ->  X
( ( t+ `  R )  u.  _I  ) a )  ->  ( Y ( t+ `  R
) M  ->  X
( ( t+ `  R )  u.  _I  ) M ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `  R ) M  ->  X ( ( t+ `  R
)  u.  _I  ) M ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1452    e. wcel 1897   _Vcvv 3056    u. cun 3413   class class class wbr 4415    _I cid 4762   `'ccnv 4851   Fun wfun 5594   ` cfv 5600   t+ctcl 13097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-frege1 36430  ax-frege2 36431  ax-frege8 36449  ax-frege52a 36497  ax-frege58b 36541
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-ifp 1436  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-fun 5602
This theorem is referenced by:  frege124  36627
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