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Theorem mptbi12f 33607
Description: Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
mptbi12f.1 𝑥𝐴
mptbi12f.2 𝑥𝐵
Assertion
Ref Expression
mptbi12f ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → (𝑥𝐴𝐷) = (𝑥𝐵𝐸))

Proof of Theorem mptbi12f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mptbi12f.1 . . . . . . . 8 𝑥𝐴
2 mptbi12f.2 . . . . . . . 8 𝑥𝐵
31, 2nfeq 2772 . . . . . . 7 𝑥 𝐴 = 𝐵
4 eleq2 2687 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
53, 4alrimi 2080 . . . . . 6 (𝐴 = 𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
6 ax-5 1836 . . . . . . 7 ((𝑥𝐴𝑥𝐵) → ∀𝑦(𝑥𝐴𝑥𝐵))
76alimi 1736 . . . . . 6 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥𝑦(𝑥𝐴𝑥𝐵))
85, 7syl 17 . . . . 5 (𝐴 = 𝐵 → ∀𝑥𝑦(𝑥𝐴𝑥𝐵))
9 eqeq2 2632 . . . . . . . . 9 (𝐷 = 𝐸 → (𝑦 = 𝐷𝑦 = 𝐸))
109alrimiv 1852 . . . . . . . 8 (𝐷 = 𝐸 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸))
1110ralimi 2947 . . . . . . 7 (∀𝑥𝐴 𝐷 = 𝐸 → ∀𝑥𝐴𝑦(𝑦 = 𝐷𝑦 = 𝐸))
12 df-ral 2912 . . . . . . 7 (∀𝑥𝐴𝑦(𝑦 = 𝐷𝑦 = 𝐸) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸)))
1311, 12sylib 208 . . . . . 6 (∀𝑥𝐴 𝐷 = 𝐸 → ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸)))
14 19.21v 1865 . . . . . . 7 (∀𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)) ↔ (𝑥𝐴 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸)))
1514albii 1744 . . . . . 6 (∀𝑥𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸)))
1613, 15sylibr 224 . . . . 5 (∀𝑥𝐴 𝐷 = 𝐸 → ∀𝑥𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))
17 id 22 . . . . . . 7 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
1817alanimi 1741 . . . . . 6 ((∀𝑦(𝑥𝐴𝑥𝐵) ∧ ∀𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ∀𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
1918alanimi 1741 . . . . 5 ((∀𝑥𝑦(𝑥𝐴𝑥𝐵) ∧ ∀𝑥𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ∀𝑥𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
208, 16, 19syl2an 494 . . . 4 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → ∀𝑥𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
21 tsan2 33581 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑥𝐴 ∨ ¬ (𝑥𝐴𝑦 = 𝐷)))
2221ord 392 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ¬ (𝑥𝐴𝑦 = 𝐷)))
23 tsbi2 33573 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸)) ∨ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))
2423ord 392 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸)) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))
2524a1dd 50 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸)) → (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))))
26 ax-1 6 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸)) → ¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))))
2725, 26contrd 33531 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸)))
2827a1d 25 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸))))
2922, 28cnf1dd 33524 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → (𝑥𝐵𝑦 = 𝐸)))
30 simplim 163 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
3130a1d 25 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ (𝑥𝐴 ∨ ¬ 𝑥𝐵) → ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
32 tsbi3 33574 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑥𝐴 ∨ ¬ 𝑥𝐵) ∨ ¬ (𝑥𝐴𝑥𝐵)))
3332ord 392 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ (𝑥𝐴 ∨ ¬ 𝑥𝐵) → ¬ (𝑥𝐴𝑥𝐵)))
34 tsan2 33581 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑥𝐴𝑥𝐵) ∨ ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
3534a1d 25 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ (𝑥𝐴 ∨ ¬ 𝑥𝐵) → ((𝑥𝐴𝑥𝐵) ∨ ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))))
3633, 35cnf1dd 33524 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ (𝑥𝐴 ∨ ¬ 𝑥𝐵) → ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
3731, 36contrd 33531 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑥𝐴 ∨ ¬ 𝑥𝐵))
3837ord 392 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ¬ 𝑥𝐵))
39 tsan2 33581 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑥𝐵 ∨ ¬ (𝑥𝐵𝑦 = 𝐸)))
4039a1d 25 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → (𝑥𝐵 ∨ ¬ (𝑥𝐵𝑦 = 𝐸))))
4138, 40cnf1dd 33524 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ¬ (𝑥𝐵𝑦 = 𝐸)))
4229, 41contrd 33531 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → 𝑥𝐴)
4342a1d 25 . . . . . . . 8 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → 𝑥𝐴))
4430a1d 25 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
45 tsan3 33582 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)) ∨ ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
4645a1d 25 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)) ∨ ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))))
4744, 46cnfn2dd 33527 . . . . . . . 8 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
4843, 47mpdd 43 . . . . . . 7 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷𝑦 = 𝐸)))
49 notnotr 125 . . . . . . . . . . . . . . . 16 (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑥𝐵𝑦 = 𝐸))
5049a1i 11 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑥𝐵𝑦 = 𝐸)))
5139a1d 25 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑥𝐵 ∨ ¬ (𝑥𝐵𝑦 = 𝐸))))
5250, 51cnfn2dd 33527 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → 𝑥𝐵))
5337a1d 25 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑥𝐴 ∨ ¬ 𝑥𝐵)))
5452, 53cnfn2dd 33527 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → 𝑥𝐴))
55 tsan3 33582 . . . . . . . . . . . . . . . 16 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑦 = 𝐸 ∨ ¬ (𝑥𝐵𝑦 = 𝐸)))
5655a1d 25 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑦 = 𝐸 ∨ ¬ (𝑥𝐵𝑦 = 𝐸))))
5750, 56cnfn2dd 33527 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → 𝑦 = 𝐸))
5830a1d 25 . . . . . . . . . . . . . . . . 17 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
5945a1d 25 . . . . . . . . . . . . . . . . 17 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ((𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)) ∨ ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))))
6058, 59cnfn2dd 33527 . . . . . . . . . . . . . . . 16 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
6154, 60mpdd 43 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑦 = 𝐷𝑦 = 𝐸)))
62 tsbi3 33574 . . . . . . . . . . . . . . . 16 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸)))
6362a1d 25 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ((𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸))))
6461, 63cnfn2dd 33527 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸)))
6557, 64cnfn2dd 33527 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → 𝑦 = 𝐷))
6654, 65jcad 555 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑥𝐴𝑦 = 𝐷)))
67 ax-1 6 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))))
68 tsim3 33571 . . . . . . . . . . . . . . . 16 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)) ∨ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))))
6968a1d 25 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (¬ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)) ∨ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))))
7067, 69cnf2dd 33525 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ¬ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))
71 tsbi1 33572 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((¬ (𝑥𝐴𝑦 = 𝐷) ∨ ¬ (𝑥𝐵𝑦 = 𝐸)) ∨ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))
7271a1d 25 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ((¬ (𝑥𝐴𝑦 = 𝐷) ∨ ¬ (𝑥𝐵𝑦 = 𝐸)) ∨ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))))
7370, 72cnf2dd 33525 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (¬ (𝑥𝐴𝑦 = 𝐷) ∨ ¬ (𝑥𝐵𝑦 = 𝐸))))
7450, 73cnfn2dd 33527 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ¬ (𝑥𝐴𝑦 = 𝐷)))
7566, 74contrd 33531 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ¬ (𝑥𝐵𝑦 = 𝐸))
7675a1d 25 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑥𝐵𝑦 = 𝐸)))
7727a1d 25 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸))))
7876, 77cnf2dd 33525 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑥𝐴𝑦 = 𝐷)))
79 tsan3 33582 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑦 = 𝐷 ∨ ¬ (𝑥𝐴𝑦 = 𝐷)))
8079a1d 25 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑦 = 𝐷 ∨ ¬ (𝑥𝐴𝑦 = 𝐷))))
8178, 80cnfn2dd 33527 . . . . . . . 8 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → 𝑦 = 𝐷))
8234a1d 25 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((𝑥𝐴𝑥𝐵) ∨ ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))))
8344, 82cnfn2dd 33527 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑥𝐴𝑥𝐵)))
84 tsbi4 33575 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((¬ 𝑥𝐴𝑥𝐵) ∨ ¬ (𝑥𝐴𝑥𝐵)))
8584a1d 25 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑥𝐴𝑥𝐵) ∨ ¬ (𝑥𝐴𝑥𝐵))))
8683, 85cnfn2dd 33527 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥𝐴𝑥𝐵)))
8743, 86cnfn1dd 33526 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → 𝑥𝐵))
88 tsan1 33580 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸) ∨ (𝑥𝐵𝑦 = 𝐸)))
8988a1d 25 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸) ∨ (𝑥𝐵𝑦 = 𝐸))))
9076, 89cnf2dd 33525 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸)))
9187, 90cnfn1dd 33526 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ¬ 𝑦 = 𝐸))
92 tsbi4 33575 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((¬ 𝑦 = 𝐷𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸)))
9392a1d 25 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑦 = 𝐷𝑦 = 𝐸) ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸))))
9493or32dd 33528 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸)) ∨ 𝑦 = 𝐸)))
9591, 94cnf2dd 33525 . . . . . . . 8 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸))))
9681, 95cnfn1dd 33526 . . . . . . 7 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑦 = 𝐷𝑦 = 𝐸)))
9748, 96contrd 33531 . . . . . 6 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ⊥)
9897efald2 33509 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))
99982alimi 1737 . . . 4 (∀𝑥𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ∀𝑥𝑦((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))
10020, 99syl 17 . . 3 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → ∀𝑥𝑦((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))
101 eqopab2b 4965 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐷)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐸)} ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))
102100, 101sylibr 224 . 2 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐷)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐸)})
103 df-mpt 4675 . 2 (𝑥𝐴𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐷)}
104 df-mpt 4675 . 2 (𝑥𝐵𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐸)}
105102, 103, 1043eqtr4g 2680 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → (𝑥𝐴𝐷) = (𝑥𝐵𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wfal 1485  wcel 1987  wnfc 2748  wral 2907  {copab 4672  cmpt 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-mpt 4675
This theorem is referenced by: (None)
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