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Theorem mptbi12f 34306
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 2914 . . . . . . 7 𝑥 𝐴 = 𝐵
4 eleq2 2828 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
53, 4alrimi 2229 . . . . . 6 (𝐴 = 𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
6 ax-5 1988 . . . . . . 7 ((𝑥𝐴𝑥𝐵) → ∀𝑦(𝑥𝐴𝑥𝐵))
76alimi 1888 . . . . . 6 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥𝑦(𝑥𝐴𝑥𝐵))
85, 7syl 17 . . . . 5 (𝐴 = 𝐵 → ∀𝑥𝑦(𝑥𝐴𝑥𝐵))
9 eqeq2 2771 . . . . . . . . 9 (𝐷 = 𝐸 → (𝑦 = 𝐷𝑦 = 𝐸))
109alrimiv 2004 . . . . . . . 8 (𝐷 = 𝐸 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸))
1110ralimi 3090 . . . . . . 7 (∀𝑥𝐴 𝐷 = 𝐸 → ∀𝑥𝐴𝑦(𝑦 = 𝐷𝑦 = 𝐸))
12 df-ral 3055 . . . . . . 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 2017 . . . . . . 7 (∀𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)) ↔ (𝑥𝐴 → ∀𝑦(𝑦 = 𝐷𝑦 = 𝐸)))
1514albii 1896 . . . . . 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 1893 . . . . . 6 ((∀𝑦(𝑥𝐴𝑥𝐵) ∧ ∀𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ∀𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
1918alanimi 1893 . . . . 5 ((∀𝑥𝑦(𝑥𝐴𝑥𝐵) ∧ ∀𝑥𝑦(𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ∀𝑥𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
208, 16, 19syl2an 495 . . . 4 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → ∀𝑥𝑦((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))))
21 tsan2 34280 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑥𝐴 ∨ ¬ (𝑥𝐴𝑦 = 𝐷)))
2221ord 391 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ¬ (𝑥𝐴𝑦 = 𝐷)))
23 tsbi2 34272 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (((𝑥𝐴𝑦 = 𝐷) ∨ (𝑥𝐵𝑦 = 𝐸)) ∨ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))
2423ord 391 . . . . . . . . . . . . . 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 34230 . . . . . . . . . . . 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 34223 . . . . . . . . . 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 34273 . . . . . . . . . . . . . . 15 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ((𝑥𝐴 ∨ ¬ 𝑥𝐵) ∨ ¬ (𝑥𝐴𝑥𝐵)))
3332ord 391 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ (𝑥𝐴 ∨ ¬ 𝑥𝐵) → ¬ (𝑥𝐴𝑥𝐵)))
34 tsan2 34280 . . . . . . . . . . . . . . 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 34223 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ (𝑥𝐴 ∨ ¬ 𝑥𝐵) → ¬ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸)))))
3731, 36contrd 34230 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (𝑥𝐴 ∨ ¬ 𝑥𝐵))
3837ord 391 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ¬ 𝑥𝐵))
39 tsan2 34280 . . . . . . . . . . . 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 34223 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ 𝑥𝐴 → ¬ (𝑥𝐵𝑦 = 𝐸)))
4229, 41contrd 34230 . . . . . . . . 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 34281 . . . . . . . . . 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 34226 . . . . . . . 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 34226 . . . . . . . . . . . . . 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 34226 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → 𝑥𝐴))
55 tsan3 34281 . . . . . . . . . . . . . . . 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 34226 . . . . . . . . . . . . . 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 34226 . . . . . . . . . . . . . . . 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 34273 . . . . . . . . . . . . . . . 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 34226 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (𝑦 = 𝐷 ∨ ¬ 𝑦 = 𝐸)))
6557, 64cnfn2dd 34226 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → 𝑦 = 𝐷))
6654, 65jcad 556 . . . . . . . . . . . 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 34270 . . . . . . . . . . . . . . . 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 34224 . . . . . . . . . . . . . 14 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ¬ ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))))
71 tsbi1 34271 . . . . . . . . . . . . . . 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 34224 . . . . . . . . . . . . 13 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → (¬ (𝑥𝐴𝑦 = 𝐷) ∨ ¬ (𝑥𝐵𝑦 = 𝐸))))
7450, 73cnfn2dd 34226 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ¬ (𝑥𝐵𝑦 = 𝐸) → ¬ (𝑥𝐴𝑦 = 𝐷)))
7566, 74contrd 34230 . . . . . . . . . . 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 34224 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑥𝐴𝑦 = 𝐷)))
79 tsan3 34281 . . . . . . . . . 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 34226 . . . . . . . 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 34226 . . . . . . . . . . . 12 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (𝑥𝐴𝑥𝐵)))
84 tsbi4 34274 . . . . . . . . . . . . 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 34226 . . . . . . . . . . 11 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥𝐴𝑥𝐵)))
8743, 86cnfn1dd 34225 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → 𝑥𝐵))
88 tsan1 34279 . . . . . . . . . . . 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 34224 . . . . . . . . . 10 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑥𝐵 ∨ ¬ 𝑦 = 𝐸)))
9187, 90cnfn1dd 34225 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ¬ 𝑦 = 𝐸))
92 tsbi4 34274 . . . . . . . . . . 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 34227 . . . . . . . . 9 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ((¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸)) ∨ 𝑦 = 𝐸)))
9591, 94cnf2dd 34224 . . . . . . . 8 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → (¬ 𝑦 = 𝐷 ∨ ¬ (𝑦 = 𝐷𝑦 = 𝐸))))
9681, 95cnfn1dd 34225 . . . . . . 7 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → (¬ ⊥ → ¬ (𝑦 = 𝐷𝑦 = 𝐸)))
9748, 96contrd 34230 . . . . . 6 (¬ (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸))) → ⊥)
9897efald2 34208 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴 → (𝑦 = 𝐷𝑦 = 𝐸))) → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐵𝑦 = 𝐸)))
99982alimi 1889 . . . 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 5155 . . 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 4882 . 2 (𝑥𝐴𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐷)}
104 df-mpt 4882 . 2 (𝑥𝐵𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐸)}
105102, 103, 1043eqtr4g 2819 1 ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → (𝑥𝐴𝐷) = (𝑥𝐵𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wal 1630   = wceq 1632  wfal 1637  wcel 2139  wnfc 2889  wral 3050  {copab 4864  cmpt 4881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-mpt 4882
This theorem is referenced by: (None)
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