tgcgrneq - Metamath Proof Explorer

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Theorem tgcgrneq 28652

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrcomlr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrcomlr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrcomlr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrcomlr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgrcomlr.6	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
tgcgrneq.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)

**Assertion**
Ref	Expression
tgcgrneq	⊢ (𝜑 → 𝐶 ≠ 𝐷)

**Proof of Theorem tgcgrneq**
Step	Hyp	Ref	Expression
1		tgcgrneq.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		tkgeom.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		tkgeom.d	. . . 4 ⊢ − = (dist‘𝐺)
4		tkgeom.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		tkgeom.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		tgcgrcomlr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7		tgcgrcomlr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8		tgcgrcomlr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9		tgcgrcomlr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10		tgcgrcomlr.6	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
11	2, 3, 4, 5, 6, 7, 8, 9, 10	tgcgreqb 28650	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
12	11	necon3bid 3001	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
13	1, 12	mpbid 234	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 distcds 17295 TarskiGcstrkg 28596 Itvcitv 28602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-ext 2734 ax-nul 5256

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-sb 2091 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3077 df-rab 3415 df-v 3456 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-iota 6477 df-fv 6529 df-ov 7399 df-trkgc 28617 df-trkg 28622

This theorem is referenced by: hlcgrex 28785 midexlem 28865 footexALT 28891 footexlem1 28892 footexlem2 28893 mideulem2 28907 opphllem3 28922 trgcopy 28977 iscgra1 29004 cgrane1 29006 cgrane2 29007 cgrcgra 29015 flatcgra 29018 cgrg3col4 29047 tgsas2 29050 tgsas3 29051 tgasa1 29052

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