Theorem iscgra1 28744

Description: A special version of iscgra 28743 where one distance is known to be equal. In this case, angle congruence can be written with only one quantifier. (Contributed by Thierry Arnoux, 9-Aug-2020.)

Hypotheses

Ref	Expression
iscgra.p	⊢ 𝑃 = (Base‘𝐺)
iscgra.i	⊢ 𝐼 = (Itv‘𝐺)
iscgra.k	⊢ 𝐾 = (hlG‘𝐺)
iscgra.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
iscgra.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
iscgra.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
iscgra.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
iscgra.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
iscgra.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
iscgra.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
iscgra1.m	⊢ − = (dist‘𝐺)
iscgra1.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
iscgra1.2	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))

Assertion

Ref	Expression
iscgra1	⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸   𝑥,𝐹   𝑥,𝐾   𝜑,𝑥   𝑥,𝐺   𝑥,𝐼   𝑥,𝑃

Allowed substitution hint:   − (𝑥)

Proof of Theorem iscgra1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscgra.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		iscgra.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		iscgra.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
4		iscgra.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		iscgra.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		iscgra.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		iscgra.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
8		iscgra.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
9		iscgra.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
10		iscgra.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	iscgra 28743	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑦 ∈ 𝑃 ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)))
12	9	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
13	6	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
14	5	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ∈ 𝑃)
15	4	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
16	8	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐷 ∈ 𝑃)
17		iscgra1.m	. . . . . . . 8 ⊢ − = (dist‘𝐺)
18		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
19		simpr2 1196	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦(𝐾‘𝐸)𝐷)
20	1, 2, 3, 18, 16, 12, 15, 19	hlne2 28540	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐷 ≠ 𝐸)
21		iscgra1.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
22	21	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ≠ 𝐵)
23	22	necomd 2981	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐵 ≠ 𝐴)
24	1, 2, 3, 16, 12, 12, 15, 20	hlid 28543	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝐷)
25		eqid 2730	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
26	7	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
27		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃)
28		simpr1 1195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
29	1, 17, 2, 25, 15, 14, 13, 26, 18, 12, 27, 28	cgr3simp1 28454	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝐴 − 𝐵) = (𝑦 − 𝐸))
30	29	eqcomd 2736	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝑦 − 𝐸) = (𝐴 − 𝐵))
31	1, 17, 2, 15, 18, 12, 14, 13, 30	tgcgrcomlr 28414	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝐸 − 𝑦) = (𝐵 − 𝐴))
32		iscgra1.2	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
33	32	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
34	33	eqcomd 2736	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝐷 − 𝐸) = (𝐴 − 𝐵))
35	1, 17, 2, 15, 16, 12, 14, 13, 34	tgcgrcomlr 28414	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝐸 − 𝐷) = (𝐵 − 𝐴))
36	1, 2, 3, 12, 13, 14, 15, 16, 17, 20, 23, 18, 16, 19, 24, 31, 35	hlcgreulem 28551	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 = 𝐷)
37		simpr3 1197	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑥(𝐾‘𝐸)𝐹)
38	36, 28, 37	jca32 515	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
39		simprrl 780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
40		simprl 770	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑦 = 𝐷)
41	8	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷 ∈ 𝑃)
42	9	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐸 ∈ 𝑃)
43	4	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐺 ∈ TarskiG)
44	1, 17, 2, 4, 5, 6, 8, 9, 32, 21	tgcgrneq 28417	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
45	44	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷 ≠ 𝐸)
46	1, 2, 3, 41, 41, 42, 43, 45	hlid 28543	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷(𝐾‘𝐸)𝐷)
47	40, 46	eqbrtrd 5132	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑦(𝐾‘𝐸)𝐷)
48		simprrr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑥(𝐾‘𝐸)𝐹)
49	39, 47, 48	3jca 1128	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹))
50	38, 49	impbida 800	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))))
51	50	rexbidva 3156	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → (∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ ∃𝑥 ∈ 𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))))
52		r19.42v 3170	. . . 4 ⊢ (∃𝑥 ∈ 𝑃 (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)) ↔ (𝑦 = 𝐷 ∧ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
53	51, 52	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → (∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ (𝑦 = 𝐷 ∧ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))))
54	53	rexbidva 3156	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝑃 ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ ∃𝑦 ∈ 𝑃 (𝑦 = 𝐷 ∧ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))))
55		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷)
56		eqidd 2731	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝐸 = 𝐸)
57		eqidd 2731	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝑥 = 𝑥)
58	55, 56, 57	s3eqd 14837	. . . . . . 7 ⊢ (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
59	58	breq2d 5122	. . . . . 6 ⊢ (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
60	59	anbi1d 631	. . . . 5 ⊢ (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
61	60	rexbidv 3158	. . . 4 ⊢ (𝑦 = 𝐷 → (∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
62	61	ceqsrexv 3624	. . 3 ⊢ (𝐷 ∈ 𝑃 → (∃𝑦 ∈ 𝑃 (𝑦 = 𝐷 ∧ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)) ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
63	8, 62	syl 17	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝑃 (𝑦 = 𝐷 ∧ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)) ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
64	11, 54, 63	3bitrd 305	1 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  ⟨“cs3 14815  Basecbs 17186  distcds 17236  TarskiGcstrkg 28361  Itvcitv 28367  cgrGccgrg 28444  hlGchlg 28534  cgrAccgra 28741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-s2 14821  df-s3 14822  df-trkgc 28382  df-trkgb 28383  df-trkgcb 28384  df-trkg 28387  df-cgrg 28445  df-hlg 28535  df-cgra 28742

This theorem is referenced by:  acopyeu  28768