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Theorem frege104 36634
Description: Proposition 104 of [Frege1879] p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis
Ref Expression
frege103.z  |-  Z  e.  V
Assertion
Ref Expression
frege104  |-  ( X ( ( t+ `  R )  u.  _I  ) Z  -> 
( -.  X ( t+ `  R
) Z  ->  X  =  Z ) )

Proof of Theorem frege104
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 frege103.z . . . 4  |-  Z  e.  V
21elexi 3041 . . 3  |-  Z  e. 
_V
3 eqeq1 2475 . . . 4  |-  ( z  =  Z  ->  (
z  =  X  <->  Z  =  X ) )
4 eqeq2 2482 . . . 4  |-  ( z  =  Z  ->  ( X  =  z  <->  X  =  Z ) )
53, 4imbi12d 327 . . 3  |-  ( z  =  Z  ->  (
( z  =  X  ->  X  =  z )  <->  ( Z  =  X  ->  X  =  Z ) ) )
6 frege55c 36585 . . 3  |-  ( z  =  X  ->  X  =  z )
72, 5, 6vtocl 3086 . 2  |-  ( Z  =  X  ->  X  =  Z )
81frege103 36633 . 2  |-  ( ( Z  =  X  ->  X  =  Z )  ->  ( X ( ( t+ `  R
)  u.  _I  ) Z  ->  ( -.  X
( t+ `  R ) Z  ->  X  =  Z )
) )
97, 8ax-mp 5 1  |-  ( X ( ( t+ `  R )  u.  _I  ) Z  -> 
( -.  X ( t+ `  R
) Z  ->  X  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904    u. cun 3388   class class class wbr 4395    _I cid 4749   ` cfv 5589   t+ctcl 13124
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-frege52c 36555
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-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
This theorem is referenced by:  frege114  36644
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