Theorem iscgra1 28836

Description: A special version of iscgra 28835 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 28835	. 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 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
13	6	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
14	5	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ∈ 𝑃)
15	4	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
16	8	ad3antrrr 729	. . . . . . . 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 1195	. . . . . . . . 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 28632	. . . . . . . 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 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ≠ 𝐵)
23	22	necomd 3002	. . . . . . . 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 28635	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝐷)
25		eqid 2740	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
26	7	ad3antrrr 729	. . . . . . . . . . 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 1194	. . . . . . . . . . 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 28546	. . . . . . . . . 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 2746	. . . . . . . . 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 28506	. . . . . . . 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 729	. . . . . . . . . 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 2746	. . . . . . . . 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 28506	. . . . . . . 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 28643	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 = 𝐷)
37		simpr3 1196	. . . . . . 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 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷 ∈ 𝑃)
42	9	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐸 ∈ 𝑃)
43	4	ad3antrrr 729	. . . . . . . . 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 28509	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
45	44	ad3antrrr 729	. . . . . . . . 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 28635	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷(𝐾‘𝐸)𝐷)
47	40, 46	eqbrtrd 5188	. . . . . . 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 3183	. . . 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 3197	. . . 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 3183	. 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 2741	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝐸 = 𝐸)
57		eqidd 2741	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝑥 = 𝑥)
58	55, 56, 57	s3eqd 14913	. . . . . . 7 ⊢ (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
59	58	breq2d 5178	. . . . . 6 ⊢ (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
60	59	anbi1d 630	. . . . 5 ⊢ (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
61	60	rexbidv 3185	. . . 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 3668	. . 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 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  ⟨“cs3 14891  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459  cgrGccgrg 28536  hlGchlg 28626  cgrAccgra 28833

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-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-cgrg 28537  df-hlg 28627  df-cgra 28834

This theorem is referenced by:  acopyeu  28860