Theorem iscgra1 27075

Description: A special version of iscgra 27074 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 27074	. 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 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
13	6	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
14	5	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ∈ 𝑃)
15	4	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
16	8	ad3antrrr 726	. . . . . . . 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 772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
19		simpr2 1193	. . . . . . . . 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 26871	. . . . . . . 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 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ≠ 𝐵)
23	22	necomd 2998	. . . . . . . 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 26874	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝐷)
25		eqid 2738	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
26	7	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
27		simplr 765	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃)
28		simpr1 1192	. . . . . . . . . . 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 26785	. . . . . . . . . 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 2744	. . . . . . . . 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 26745	. . . . . . . 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 726	. . . . . . . . . 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 2744	. . . . . . . . 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 26745	. . . . . . . 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 26882	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 = 𝐷)
37		simpr3 1194	. . . . . . 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 777	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
40		simprl 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑦 = 𝐷)
41	8	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷 ∈ 𝑃)
42	9	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐸 ∈ 𝑃)
43	4	ad3antrrr 726	. . . . . . . . 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 26748	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
45	44	ad3antrrr 726	. . . . . . . . 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 26874	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷(𝐾‘𝐸)𝐷)
47	40, 46	eqbrtrd 5092	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑦(𝐾‘𝐸)𝐷)
48		simprrr 778	. . . . . . 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 1126	. . . . . 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 797	. . . . 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 3224	. . . 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 3276	. . . 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 286	. . 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 3224	. 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 2739	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝐸 = 𝐸)
57		eqidd 2739	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝑥 = 𝑥)
58	55, 56, 57	s3eqd 14505	. . . . . . 7 ⊢ (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
59	58	breq2d 5082	. . . . . 6 ⊢ (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
60	59	anbi1d 629	. . . . 5 ⊢ (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
61	60	rexbidv 3225	. . . 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 3578	. . 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 304	1 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  ⟨“cs3 14483  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699  cgrGccgrg 26775  hlGchlg 26865  cgrAccgra 27072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-trkgc 26713  df-trkgb 26714  df-trkgcb 26715  df-trkg 26718  df-cgrg 26776  df-hlg 26866  df-cgra 27073

This theorem is referenced by:  acopyeu  27099