Theorem frege55c 36558

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 3059	. . . 4 |- x e. _V
2	1	frege54cor1c 36555	. . 3 |- [. x / y ]. y = x
3		frege53c 36554	. . 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 3279	. . . 4 |- ( [. A / y ]. y = x <-> A e. { y | y = x } )
6		clelab 2586	. . . 4 |- ( A e. { y | y = x } <-> E. y ( y = A /\ y = x ) )
7	5, 6	bitri 257	. . 3 |- ( [. A / y ]. y = x <-> E. y ( y = A /\ y = x ) )
8		eqtr2 2481	. . . 4 |- ( ( y = A /\ y = x ) -> A = x )
9	8	exlimiv 1786	. . 3 |- ( E. y ( y = A /\ y = x ) -> A = x )
10	7, 9	sylbi 200	. 2 |- ( [. A / y ]. y = x -> A = x )
11	4, 10	syl 17	1 |- ( x = A -> A = x )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   E.wex 1673   e. wcel 1897   {cab 2447   _Vcvv 3056   [.wsbc 3278

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-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-frege8 36449  ax-frege52c 36528

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-sbc 3279  df-sn 3980

This theorem is referenced by:  frege104  36607