mptbi12f - Mathbox for Giovanni Mascellani

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Theorem mptbi12f 33145

Description: Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

**Hypotheses**
Ref	Expression
mptbi12f.1	⊢ Ⅎ𝑥𝐴
mptbi12f.2	⊢ Ⅎ𝑥𝐵

**Assertion**
Ref	Expression
mptbi12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸))

Proof of Theorem mptbi12f Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptbi12f.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
2		mptbi12f.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2762	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2677	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	3, 4	alrimi 2069	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
6		ax-5 1827	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
7	6	alimi 1730	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
8	5, 7	syl 17	. . . . 5 ⊢ (𝐴 = 𝐵 → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
9		eqeq2 2621	. . . . . . . . 9 ⊢ (𝐷 = 𝐸 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))
10	9	alrimiv 1842	. . . . . . . 8 ⊢ (𝐷 = 𝐸 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))
11	10	ralimi 2936	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))
12		df-ral 2901	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
13	11, 12	sylib 207	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
14		19.21v 1855	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
15	14	albii 1737	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
16	13, 15	sylibr 223	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
17		id 22	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
18	17	alanimi 1734	. . . . . 6 ⊢ ((∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
19	18	alanimi 1734	. . . . 5 ⊢ ((∀𝑥∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
20	8, 16, 19	syl2an 493	. . . 4 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
21		tsan2 33119	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑥 ∈ 𝐴 ∨ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
22	21	ord 391	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
23		tsbi2 33111	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
24	23	ord 391	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
25	24	a1dd 48	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) → (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))))
26		ax-1 6	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) → ¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))))
27	25, 26	contrd 33069	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
28	27	a1d 25	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
29	22, 28	cnf1dd 33062	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
30		simplim 162	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
31	30	a1d 25	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))))
32		tsbi3 33112	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵) ∨ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
33	32	ord 391	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵) → ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
34		tsan2 33119	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∨ ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))))
35	34	a1d 25	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∨ ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))))
36	33, 35	cnf1dd 33062	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵) → ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))))
37	31, 36	contrd 33069	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵))
38	37	ord 391	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
39		tsan2 33119	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑥 ∈ 𝐵 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
40	39	a1d 25	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
41	38, 40	cnf1dd 33062	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
42	29, 41	contrd 33069	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → 𝑥 ∈ 𝐴)
43	42	a1d 25	. . . . . . . 8 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑥 ∈ 𝐴))
44	30	a1d 25	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))))
45		tsan3 33120	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ∨ ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))))
46	45	a1d 25	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ∨ ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))))
47	44, 46	cnfn2dd 33065	. . . . . . . 8 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
48	43, 47	mpdd 42	. . . . . . 7 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
49		notnotr 124	. . . . . . . . . . . . . . . 16 ⊢ (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))
50	49	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
51	39	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐵 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
52	50, 51	cnfn2dd 33065	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑥 ∈ 𝐵))
53	37	a1d 25	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵)))
54	52, 53	cnfn2dd 33065	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑥 ∈ 𝐴))
55		tsan3 33120	. . . . . . . . . . . . . . . 16 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑦 = 𝐸 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
56	55	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑦 = 𝐸 ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
57	50, 56	cnfn2dd 33065	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑦 = 𝐸))
58	30	a1d 25	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))))
59	45	a1d 25	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ((𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ∨ ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))))
60	58, 59	cnfn2dd 33065	. . . . . . . . . . . . . . . 16 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
61	54, 60	mpdd 42	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
62		tsbi3 33112	. . . . . . . . . . . . . . . 16 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
63	62	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ((𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
64	61, 63	cnfn2dd 33065	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸)))
65	57, 64	cnfn2dd 33065	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → 𝑦 = 𝐷))
66	54, 65	jcad 554	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
67		ax-1 6	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))))
68		tsim3 33109	. . . . . . . . . . . . . . . 16 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) ∨ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))))
69	68	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) ∨ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))))
70	67, 69	cnf2dd 33063	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
71		tsbi1 33110	. . . . . . . . . . . . . . 15 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
72	71	a1d 25	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ((¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)) ∨ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))))
73	70, 72	cnf2dd 33063	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
74	50, 73	cnfn2dd 33065	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
75	66, 74	contrd 33069	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))
76	75	a1d 25	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
77	27	a1d 25	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
78	76, 77	cnf2dd 33063	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
79		tsan3 33120	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑦 = 𝐷 ∨ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
80	79	a1d 25	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷 ∨ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷))))
81	78, 80	cnfn2dd 33065	. . . . . . . 8 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑦 = 𝐷))
82	34	a1d 25	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∨ ¬ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))))
83	44, 82	cnfn2dd 33065	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
84		tsbi4 33113	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∨ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
85	84	a1d 25	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∨ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))))
86	83, 85	cnfn2dd 33065	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
87	43, 86	cnfn1dd 33064	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑥 ∈ 𝐵))
88		tsan1 33118	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
89	88	a1d 25	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸) ∨ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))))
90	76, 89	cnf2dd 33063	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸)))
91	87, 90	cnfn1dd 33064	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ 𝑦 = 𝐸))
92		tsbi4 33113	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ((¬ 𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
93	92	a1d 25	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
94	93	or32dd 33066	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ∨ 𝑦 = 𝐸)))
95	91, 94	cnf2dd 33063	. . . . . . . 8 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
96	81, 95	cnfn1dd 33064	. . . . . . 7 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
97	48, 96	contrd 33069	. . . . . 6 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ⊥)
98	97	efald2 33047	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
99	98	2alimi 1731	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
100	20, 99	syl 17	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
101		eqopab2b 4930	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)} ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
102	100, 101	sylibr 223	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)})
103		df-mpt 4645	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)}
104		df-mpt 4645	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)}
105	102, 103, 104	3eqtr4g 2669	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∀wal 1473 = wceq 1475 ⊥wfal 1480 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 {copab 4642 ↦ cmpt 4643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-mpt 4645

This theorem is referenced by: (None)

W3C validator