tgcgrneq - Metamath Proof Explorer

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Theorem tgcgrneq 28509

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 28507	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
12	11	necon3bid 2991	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))
13	1, 12	mpbid 232	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 distcds 17320 TarskiGcstrkg 28453 Itvcitv 28459

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-nul 5324

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-iota 6525 df-fv 6581 df-ov 7451 df-trkgc 28474 df-trkg 28479

This theorem is referenced by: hlcgrex 28642 midexlem 28718 footexALT 28744 footexlem1 28745 footexlem2 28746 mideulem2 28760 opphllem3 28775 trgcopy 28830 iscgra1 28836 cgrane1 28838 cgrane2 28839 cgrcgra 28847 flatcgra 28850 cgrg3col4 28879 tgsas2 28882 tgsas3 28883 tgasa1 28884

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