Theorem fidifsnen 6860

Description: All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)

Assertion

Ref	Expression
fidifsnen	⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

Proof of Theorem fidifsnen

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difexg 4139	. . . . . 6 ⊢ (𝑋 ∈ Fin → (𝑋 ∖ {𝐴}) ∈ V)
2	1	3ad2ant1 1018	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ∈ V)
3	2	adantr 276	. . . 4 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
4		enrefg 6754	. . . 4 ⊢ ((𝑋 ∖ {𝐴}) ∈ V → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
5	3, 4	syl 14	. . 3 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐴}))
6		sneq 3600	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
7	6	difeq2d 3251	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
8	7	adantl 277	. . 3 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) = (𝑋 ∖ {𝐵}))
9	5, 8	breqtrd 4024	. 2 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
10	2	adantr 276	. . 3 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ∈ V)
11		eqid 2175	. . . 4 ⊢ (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) = (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
12		iftrue 3537	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
13	12	adantl 277	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
14		simpll2 1037	. . . . . . . 8 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐴 ∈ 𝑋)
15	14	adantr 276	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → 𝐴 ∈ 𝑋)
16	13, 15	eqeltrd 2252	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
17		simpllr 534	. . . . . . . 8 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ 𝐴 = 𝐵)
18	13	eqeq1d 2184	. . . . . . . 8 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵 ↔ 𝐴 = 𝐵))
19	17, 18	mtbird 673	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
20	19	neneqad 2424	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
21		eldifsn 3716	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}) ↔ (if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵))
22	16, 20, 21	sylanbrc 417	. . . . 5 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
23		iffalse 3540	. . . . . . . 8 ⊢ (¬ 𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
24	23	adantl 277	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
25		eldifi 3255	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑋 ∖ {𝐴}) → 𝑥 ∈ 𝑋)
26	25	ad2antlr 489	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝑋)
27	24, 26	eqeltrd 2252	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ 𝑋)
28		simpr 110	. . . . . . . 8 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵)
29	24	eqeq1d 2184	. . . . . . . 8 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → (if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵 ↔ 𝑥 = 𝐵))
30	28, 29	mtbird 673	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → ¬ if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐵)
31	30	neneqad 2424	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ≠ 𝐵)
32	27, 31, 21	sylanbrc 417	. . . . 5 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
33		simpll1 1036	. . . . . . 7 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑋 ∈ Fin)
34	25	adantl 277	. . . . . . 7 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝑥 ∈ 𝑋)
35		simpll3 1038	. . . . . . 7 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → 𝐵 ∈ 𝑋)
36		fidceq 6859	. . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → DECID 𝑥 = 𝐵)
37	33, 34, 35, 36	syl3anc 1238	. . . . . 6 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → DECID 𝑥 = 𝐵)
38		exmiddc 836	. . . . . 6 ⊢ (DECID 𝑥 = 𝐵 → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
39	37, 38	syl 14	. . . . 5 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
40	22, 32, 39	mpjaodan 798	. . . 4 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑥 ∈ (𝑋 ∖ {𝐴})) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝑋 ∖ {𝐵}))
41		iftrue 3537	. . . . . . 7 ⊢ (𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
42	41	adantl 277	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
43		simpl3 1002	. . . . . . . 8 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑋)
44		simpr 110	. . . . . . . . . 10 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
45	44	neneqad 2424	. . . . . . . . 9 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵)
46	45	necomd 2431	. . . . . . . 8 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ≠ 𝐴)
47		eldifsn 3716	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑋 ∖ {𝐴}) ↔ (𝐵 ∈ 𝑋 ∧ 𝐵 ≠ 𝐴))
48	43, 46, 47	sylanbrc 417	. . . . . . 7 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
49	48	ad2antrr 488	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → 𝐵 ∈ (𝑋 ∖ {𝐴}))
50	42, 49	eqeltrd 2252	. . . . 5 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
51		iffalse 3540	. . . . . . 7 ⊢ (¬ 𝑦 = 𝐴 → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
52	51	adantl 277	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
53		eldifi 3255	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐵}) → 𝑦 ∈ 𝑋)
54	53	ad2antlr 489	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ 𝑋)
55		simpr 110	. . . . . . . 8 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 = 𝐴)
56	55	neneqad 2424	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ≠ 𝐴)
57		eldifsn 3716	. . . . . . 7 ⊢ (𝑦 ∈ (𝑋 ∖ {𝐴}) ↔ (𝑦 ∈ 𝑋 ∧ 𝑦 ≠ 𝐴))
58	54, 56, 57	sylanbrc 417	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐴}))
59	52, 58	eqeltrd 2252	. . . . 5 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
60		simpll1 1036	. . . . . . 7 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑋 ∈ Fin)
61	53	adantl 277	. . . . . . 7 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝑦 ∈ 𝑋)
62		simpll2 1037	. . . . . . 7 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → 𝐴 ∈ 𝑋)
63		fidceq 6859	. . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → DECID 𝑦 = 𝐴)
64	60, 61, 62, 63	syl3anc 1238	. . . . . 6 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → DECID 𝑦 = 𝐴)
65		exmiddc 836	. . . . . 6 ⊢ (DECID 𝑦 = 𝐴 → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
66	64, 65	syl 14	. . . . 5 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
67	50, 59, 66	mpjaodan 798	. . . 4 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵})) → if(𝑦 = 𝐴, 𝐵, 𝑦) ∈ (𝑋 ∖ {𝐴}))
68	12	adantl 277	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝐴)
69	68	eqeq2d 2187	. . . . . . . . 9 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝐴))
70	69	biimpar 297	. . . . . . . 8 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ 𝑦 = 𝐴) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
71	70	a1d 22	. . . . . . 7 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ 𝑦 = 𝐴) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
72		simpr 110	. . . . . . . . . . 11 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
73	51	eqeq2d 2187	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
74	73	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝑦))
75	72, 74	mpbid 147	. . . . . . . . . 10 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑥 = 𝑦)
76		simpllr 534	. . . . . . . . . 10 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑥 = 𝐵)
77	75, 76	eqtr3d 2210	. . . . . . . . 9 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑦 = 𝐵)
78		simprr 531	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
79	78	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → 𝑦 ∈ (𝑋 ∖ {𝐵}))
80	79	eldifbd 3139	. . . . . . . . . . 11 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → ¬ 𝑦 ∈ {𝐵})
81	80	adantr 276	. . . . . . . . . 10 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → ¬ 𝑦 ∈ {𝐵})
82		velsn 3606	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
83	81, 82	sylnib 676	. . . . . . . . 9 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → ¬ 𝑦 = 𝐵)
84	77, 83	pm2.21dd 620	. . . . . . . 8 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
85	84	ex 115	. . . . . . 7 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) ∧ ¬ 𝑦 = 𝐴) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
86		simpll1 1036	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑋 ∈ Fin)
87	53	ad2antll 491	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑦 ∈ 𝑋)
88		simpll2 1037	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝐴 ∈ 𝑋)
89	86, 87, 88, 63	syl3anc 1238	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → DECID 𝑦 = 𝐴)
90	89, 65	syl 14	. . . . . . . 8 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
91	90	adantr 276	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
92	71, 85, 91	mpjaodan 798	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
93	41	eqeq2d 2187	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑥 = 𝐵))
94	93	biimprcd 160	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐴 → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
95	94	adantl 277	. . . . . . 7 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = 𝐴 → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
96	69, 95	sylbid 150	. . . . . 6 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)))
97	92, 96	impbid 129	. . . . 5 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ 𝑥 = 𝐵) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
98		simplr 528	. . . . . . . . 9 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
99	41	adantl 277	. . . . . . . . 9 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝐵)
100	98, 99	eqtrd 2208	. . . . . . . 8 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → 𝑥 = 𝐵)
101		simpllr 534	. . . . . . . 8 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → ¬ 𝑥 = 𝐵)
102	100, 101	pm2.21dd 620	. . . . . . 7 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ 𝑦 = 𝐴) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
103	23	ad3antlr 493	. . . . . . . 8 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → if(𝑥 = 𝐵, 𝐴, 𝑥) = 𝑥)
104		simplr 528	. . . . . . . . 9 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
105	51	adantl 277	. . . . . . . . 9 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
106	104, 105	eqtrd 2208	. . . . . . . 8 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → 𝑥 = 𝑦)
107	103, 106	eqtr2d 2209	. . . . . . 7 ⊢ (((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) ∧ ¬ 𝑦 = 𝐴) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
108	90	ad2antrr 488	. . . . . . 7 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → (𝑦 = 𝐴 ∨ ¬ 𝑦 = 𝐴))
109	102, 107, 108	mpjaodan 798	. . . . . 6 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦)) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
110		simprl 529	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → 𝑥 ∈ (𝑋 ∖ {𝐴}))
111	110	eldifbd 3139	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 ∈ {𝐴})
112		velsn 3606	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
113	111, 112	sylnib 676	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → ¬ 𝑥 = 𝐴)
114	113	ad2antrr 488	. . . . . . . . 9 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑥 = 𝐴)
115		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥))
116	23	eqeq2d 2187	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝐵 → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
117	116	ad2antlr 489	. . . . . . . . . . 11 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → (𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥) ↔ 𝑦 = 𝑥))
118	115, 117	mpbid 147	. . . . . . . . . 10 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑦 = 𝑥)
119	118	eqeq1d 2184	. . . . . . . . 9 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
120	114, 119	mtbird 673	. . . . . . . 8 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → ¬ 𝑦 = 𝐴)
121	120, 51	syl 14	. . . . . . 7 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → if(𝑦 = 𝐴, 𝐵, 𝑦) = 𝑦)
122	121, 118	eqtr2d 2209	. . . . . 6 ⊢ ((((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) ∧ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)) → 𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦))
123	109, 122	impbida 596	. . . . 5 ⊢ (((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
124	39	adantrr 479	. . . . 5 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵))
125	97, 123, 124	mpjaodan 798	. . . 4 ⊢ ((((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ∧ 𝑦 ∈ (𝑋 ∖ {𝐵}))) → (𝑥 = if(𝑦 = 𝐴, 𝐵, 𝑦) ↔ 𝑦 = if(𝑥 = 𝐵, 𝐴, 𝑥)))
126	11, 40, 67, 125	f1o2d 6066	. . 3 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵}))
127		f1oeng 6747	. . 3 ⊢ (((𝑋 ∖ {𝐴}) ∈ V ∧ (𝑥 ∈ (𝑋 ∖ {𝐴}) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)):(𝑋 ∖ {𝐴})–1-1-onto→(𝑋 ∖ {𝐵})) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
128	10, 126, 127	syl2anc 411	. 2 ⊢ (((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ¬ 𝐴 = 𝐵) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
129		fidceq 6859	. . 3 ⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → DECID 𝐴 = 𝐵)
130		exmiddc 836	. . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
131	129, 130	syl 14	. 2 ⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
132	9, 128, 131	mpjaodan 798	1 ⊢ ((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   ∧ w3a 978   = wceq 1353   ∈ wcel 2146   ≠ wne 2345  Vcvv 2735   ∖ cdif 3124  ifcif 3532  {csn 3589   class class class wbr 3998   ↦ cmpt 4059  –1-1-onto→wf1o 5207   ≈ cen 6728  Fincfn 6730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-en 6731  df-fin 6733

This theorem is referenced by:  dif1en  6869