mptbi12f - Mathbox for Giovanni Mascellani

	Mathbox for Giovanni Mascellani		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mptbi12f		Structured version Visualization version GIF version

Theorem mptbi12f 39066

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 2936	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2850	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	3, 4	alrimi 2250	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
6		ax-5 1943	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) → ∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
7	5, 6	sylg 1856	. . . . 5 ⊢ (𝐴 = 𝐵 → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
8		eqeq2 2773	. . . . . . . . 9 ⊢ (𝐷 = 𝐸 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))
9	8	alrimiv 1960	. . . . . . . 8 ⊢ (𝐷 = 𝐸 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))
10	9	ralimi 3100	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸))
11		df-ral 3078	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
12	10, 11	sylib 221	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
13		19.21v 1972	. . . . . . 7 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
14	13	albii 1852	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
15	12, 14	sylibr 237	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐷 = 𝐸 → ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
16		id 23	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
17	16	alanimi 1849	. . . . . 6 ⊢ ((∀𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑦(𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
18	17	alanimi 1849	. . . . 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))))
19	7, 15, 18	syl2an 608	. . . 4 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))))
20		tsan2 39042	. . . . . . . . . . . 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)))
21	20	ord 878	. . . . . . . . . . 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)))
22		tsbi2 39034	. . . . . . . . . . . . . . 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))))
23	22	ord 878	. . . . . . . . . . . . . 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))))
24	23	a1dd 51	. . . . . . . . . . . . 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)))))
25		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)))))
26	24, 25	contrd 38997	. . . . . . . . . . . 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)))
27	26	a1d 26	. . . . . . . . . . 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))))
28	21, 27	cnf1dd 38990	. . . . . . . . . 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)))
29		simplim 168	. . . . . . . . . . . . . 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))))
30	29	a1d 26	. . . . . . . . . . . . 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)))))
31		tsbi3 39035	. . . . . . . . . . . . . . 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)))
32	31	ord 878	. . . . . . . . . . . . . 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)))
33		tsan2 39042	. . . . . . . . . . . . . . 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)))))
34	33	a1d 26	. . . . . . . . . . . . . 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))))))
35	32, 34	cnf1dd 38990	. . . . . . . . . . . . 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)))))
36	30, 35	contrd 38997	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐵))
37	36	ord 878	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
38		tsan2 39042	. . . . . . . . . . . 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)))
39	38	a1d 26	. . . . . . . . . . 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))))
40	37, 39	cnf1dd 38990	. . . . . . . . . 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)))
41	28, 40	contrd 38997	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → 𝑥 ∈ 𝐴)
42	41	a1d 26	. . . . . . . 8 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑥 ∈ 𝐴))
43	29	a1d 26	. . . . . . . . 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)))))
44		tsan3 39043	. . . . . . . . . 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)))))
45	44	a1d 26	. . . . . . . . 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))))))
46	43, 45	cnfn2dd 38993	. . . . . . . 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))))
47	42, 46	mpdd 44	. . . . . . 7 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
48		notnotr 131	. . . . . . . . . . . . . . . 16 ⊢ (¬ ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸) → (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))
49	48	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)))
50	38	a1d 26	. . . . . . . . . . . . . . 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))))
51	49, 50	cnfn2dd 38993	. . . . . . . . . . . . . 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))
52	36	a1d 26	. . . . . . . . . . . . . 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)))
53	51, 52	cnfn2dd 38993	. . . . . . . . . . . . 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))
54		tsan3 39043	. . . . . . . . . . . . . . . 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)))
55	54	a1d 26	. . . . . . . . . . . . . . 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))))
56	49, 55	cnfn2dd 38993	. . . . . . . . . . . . . 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))
57	29	a1d 26	. . . . . . . . . . . . . . . . 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)))))
58	44	a1d 26	. . . . . . . . . . . . . . . . 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))))))
59	57, 58	cnfn2dd 38993	. . . . . . . . . . . . . . . 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))))
60	53, 59	mpdd 44	. . . . . . . . . . . . . . 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)))
61		tsbi3 39035	. . . . . . . . . . . . . . . 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)))
62	61	a1d 26	. . . . . . . . . . . . . . 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))))
63	60, 62	cnfn2dd 38993	. . . . . . . . . . . . . 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)))
64	56, 63	cnfn2dd 38993	. . . . . . . . . . . . 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))
65	53, 64	jcad 522	. . . . . . . . . . . 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)))
66		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)))))
67		tsim3 39032	. . . . . . . . . . . . . . . 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)))))
68	67	a1d 26	. . . . . . . . . . . . . . 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))))))
69	66, 68	cnf2dd 38991	. . . . . . . . . . . . . 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))))
70		tsbi1 39033	. . . . . . . . . . . . . . 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))))
71	70	a1d 26	. . . . . . . . . . . . . 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)))))
72	69, 71	cnf2dd 38991	. . . . . . . . . . . . 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))))
73	49, 72	cnfn2dd 38993	. . . . . . . . . . . 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)))
74	65, 73	contrd 38997	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))
75	74	a1d 26	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
76	26	a1d 26	. . . . . . . . . 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))))
77	75, 76	cnf2dd 38991	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)))
78		tsan3 39043	. . . . . . . . . 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)))
79	78	a1d 26	. . . . . . . . 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))))
80	77, 79	cnfn2dd 38993	. . . . . . . 8 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑦 = 𝐷))
81	33	a1d 26	. . . . . . . . . . . . 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))))))
82	43, 81	cnfn2dd 38993	. . . . . . . . . . . 12 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
83		tsbi4 39036	. . . . . . . . . . . . 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)))
84	83	a1d 26	. . . . . . . . . . . 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))))
85	82, 84	cnfn2dd 38993	. . . . . . . . . . 11 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
86	42, 85	cnfn1dd 38992	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → 𝑥 ∈ 𝐵))
87		tsan1 39041	. . . . . . . . . . . 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)))
88	87	a1d 26	. . . . . . . . . . 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))))
89	75, 88	cnf2dd 38991	. . . . . . . . . 10 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑦 = 𝐸)))
90	86, 89	cnfn1dd 38992	. . . . . . . . 9 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ 𝑦 = 𝐸))
91		tsbi4 39036	. . . . . . . . . . 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)))
92	91	a1d 26	. . . . . . . . . 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))))
93	92	or32dd 38994	. . . . . . . . 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)))
94	90, 93	cnf2dd 38991	. . . . . . . 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))))
95	80, 94	cnfn1dd 38992	. . . . . . 7 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑦 = 𝐷 ↔ 𝑦 = 𝐸)))
96	47, 95	contrd 38997	. . . . . 6 ⊢ (¬ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸))) → ⊥)
97	96	efald2 38980	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
98	97	2alimi 1845	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → (𝑦 = 𝐷 ↔ 𝑦 = 𝐸))) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
99	19, 98	syl 18	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
100		eqopab2bw 5523	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)} ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)))
101	99, 100	sylibr 237	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)})
102		df-mpt 5187	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷)}
103		df-mpt 5187	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐸)}
104	101, 102, 103	3eqtr4g 2821	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∀wal 1568 = wceq 1570 ⊥wfal 1582 ∈ wcel 2145 Ⅎwnfc 2908 ∀wral 3077 {copab 5167 ↦ cmpt 5186

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 2213 ax-ext 2733 ax-sep 5249 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ral 3078 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-opab 5168 df-mpt 5187

This theorem is used by: (None)

W3C validator