Theorem frege55c 38212

Description: Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55c	|- ( x = A -> A = x )

Proof of Theorem frege55c

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- x e. _V
2	1	frege54cor1c 38209	. . 3 |- [. x / y ]. y = x
3		frege53c 38208	. . 3 |- ( [. x / y ]. y = x -> ( x = A -> [. A / y ]. y = x ) )
4	2, 3	ax-mp 5	. 2 |- ( x = A -> [. A / y ]. y = x )
5		df-sbc 3436	. . . 4 |- ( [. A / y ]. y = x <-> A e. { y | y = x } )
6		clelab 2748	. . . 4 |- ( A e. { y | y = x } <-> E. y ( y = A /\ y = x ) )
7	5, 6	bitri 264	. . 3 |- ( [. A / y ]. y = x <-> E. y ( y = A /\ y = x ) )
8		eqtr2 2642	. . . 4 |- ( ( y = A /\ y = x ) -> A = x )
9	8	exlimiv 1858	. . 3 |- ( E. y ( y = A /\ y = x ) -> A = x )
10	7, 9	sylbi 207	. 2 |- ( [. A / y ]. y = x -> A = x )
11	4, 10	syl 17	1 |- ( x = A -> A = x )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435

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-frege8 38103  ax-frege52c 38182

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-sn 4178

This theorem is referenced by:  frege104  38261