Theorem iscgra1 29204

Description: A special version of iscgra 29203 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 29203	. 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 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃)
13	6	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐵 ∈ 𝑃)
14	5	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ∈ 𝑃)
15	4	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG)
16	8	ad3antrrr 743	. . . . . . . 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 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃)
19		simpr2 1214	. . . . . . . . 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 28959	. . . . . . . 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 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐴 ≠ 𝐵)
23	22	necomd 3012	. . . . . . . 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 28962	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝐷)
25		eqid 2762	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
26	7	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝐶 ∈ 𝑃)
27		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃)
28		simpr1 1213	. . . . . . . . . . 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 28870	. . . . . . . . . 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 2768	. . . . . . . . 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 28829	. . . . . . . 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 743	. . . . . . . . . 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 2768	. . . . . . . . 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 28829	. . . . . . . 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 28970	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑦(𝐾‘𝐸)𝐷 ∧ 𝑥(𝐾‘𝐸)𝐹)) → 𝑦 = 𝐷)
37		simpr3 1215	. . . . . . 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 525	. . . . . 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 793	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩)
40		simprl 783	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑦 = 𝐷)
41	8	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷 ∈ 𝑃)
42	9	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐸 ∈ 𝑃)
43	4	ad3antrrr 743	. . . . . . . . 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 28832	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
45	44	ad3antrrr 743	. . . . . . . . 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 28962	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝐷(𝐾‘𝐸)𝐷)
47	40, 46	eqbrtrd 5131	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ∈ 𝑃) ∧ (𝑦 = 𝐷 ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹))) → 𝑦(𝐾‘𝐸)𝐷)
48		simprrr 794	. . . . . . 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 1146	. . . . . 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 813	. . . . 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 3186	. . . 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 3196	. . . 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 290	. . 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 3186	. 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 23	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝑦 = 𝐷)
56		eqidd 2763	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝐸 = 𝐸)
57		eqidd 2763	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → 𝑥 = 𝑥)
58	55, 56, 57	s3eqd 14939	. . . . . . 7 ⊢ (𝑦 = 𝐷 → ⟨“𝑦𝐸𝑥”⟩ = ⟨“𝐷𝐸𝑥”⟩)
59	58	breq2d 5119	. . . . . 6 ⊢ (𝑦 = 𝐷 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩))
60	59	anbi1d 643	. . . . 5 ⊢ (𝑦 = 𝐷 → ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑦𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹) ↔ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))
61	60	rexbidv 3188	. . . 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 3612	. . 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 18	. 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 308	1 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝑥”⟩ ∧ 𝑥(𝐾‘𝐸)𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  ⟨“cs3 14917  Basecbs 17307  distcds 17357  TarskiGcstrkg 28776  Itvcitv 28782  cgrGccgrg 28860  hlGchlg 28950  cgrAccgra 29201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-dju 9910  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-xnn0 12606  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-hash 14399  df-word 14583  df-concat 14640  df-s1 14667  df-s2 14923  df-s3 14924  df-trkgc 28797  df-trkgb 28798  df-trkgcb 28799  df-trkg 28802  df-cgrg 28861  df-hlg 28951  df-cgra 29202

This theorem is used by:  acopyeu  29229