Theorem frege105 38262

Description: Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege103.z	|- Z e. V

Assertion

Ref	Expression
frege105	|- ( ( -. X ( t+ ` R ) Z -> Z = X ) -> X ( ( t+ ` R ) u. _I ) Z )

Proof of Theorem frege105

Step	Hyp	Ref	Expression
1		frege103.z	. . 3 |- Z e. V
2	1	dffrege99 38256	. 2 |- ( ( -. X ( t+ ` R ) Z -> Z = X ) <-> X ( ( t+ ` R ) u. _I ) Z )
3		frege52aid 38152	. 2 |- ( ( ( -. X ( t+ ` R ) Z -> Z = X ) <-> X ( ( t+ ` R ) u. _I ) Z ) -> ( ( -. X ( t+ ` R ) Z -> Z = X ) -> X ( ( t+ ` R ) u. _I ) Z ) )
4	2, 3	ax-mp 5	1 |- ( ( -. X ( t+ ` R ) Z -> Z = X ) -> X ( ( t+ ` R ) u. _I ) Z )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = 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-frege52a 38151

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-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:  frege106  38263  frege112  38269