Theorem tgcgreq 25377

Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgcgrcomlr.a	|- ( ph -> A e. P )
tgcgrcomlr.b	|- ( ph -> B e. P )
tgcgrcomlr.c	|- ( ph -> C e. P )
tgcgrcomlr.d	|- ( ph -> D e. P )
tgcgrcomlr.6	|- ( ph -> ( A .- B ) = ( C .- D ) )
tgcgreq.1	|- ( ph -> A = B )

Assertion

Ref	Expression
tgcgreq	|- ( ph -> C = D )

Proof of Theorem tgcgreq

Step	Hyp	Ref	Expression
1		tgcgreq.1	. 2 |- ( ph -> A = B )
2		tkgeom.p	. . 3 |- P = ( Base ` G )
3		tkgeom.d	. . 3 |- .- = ( dist ` G )
4		tkgeom.i	. . 3 |- I = (Itv ` G )
5		tkgeom.g	. . 3 |- ( ph -> G e. TarskiG )
6		tgcgrcomlr.a	. . 3 |- ( ph -> A e. P )
7		tgcgrcomlr.b	. . 3 |- ( ph -> B e. P )
8		tgcgrcomlr.c	. . 3 |- ( ph -> C e. P )
9		tgcgrcomlr.d	. . 3 |- ( ph -> D e. P )
10		tgcgrcomlr.6	. . 3 |- ( ph -> ( A .- B ) = ( C .- D ) )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	tgcgreqb 25376	. 2 |- ( ph -> ( A = B <-> C = D ) )
12	1, 11	mpbid 222	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

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-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgc 25347  df-trkg 25352

This theorem is referenced by:  tgcgrextend  25380  tgidinside  25466  tgbtwnconn1lem3  25469  krippenlem  25585  ragcgr  25602  lmiisolem  25688  cgrg3col4  25734