Theorem frege104 38261

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.

Step	Hyp	Ref	Expression
1		frege103.z	. . . 4 |- Z e. V
2	1	elexi 3213	. . 3 |- Z e. _V
3		eqeq1 2626	. . . 4 |- ( z = Z -> ( z = X <-> Z = X ) )
4		eqeq2 2633	. . . 4 |- ( z = Z -> ( X = z <-> X = Z ) )
5	3, 4	imbi12d 334	. . 3 |- ( z = Z -> ( ( z = X -> X = z ) <-> ( Z = X -> X = Z ) ) )
6		frege55c 38212	. . 3 |- ( z = X -> X = z )
7	2, 5, 6	vtocl 3259	. 2 |- ( Z = X -> X = Z )
8	1	frege103 38260	. 2 |- ( ( Z = X -> X = Z ) -> ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> X = Z ) ) )
9	7, 8	ax-mp 5	1 |- ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> X = Z ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   class class class wbr 4653   _I cid 5023   ` cfv 5888   t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52a 38151  ax-frege52c 38182

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121

This theorem is referenced by:  frege114  38271