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Theorem propeqop 4935
Description: Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)
Hypotheses
Ref Expression
snopeqop.a 𝐴 ∈ V
snopeqop.b 𝐵 ∈ V
snopeqop.c 𝐶 ∈ V
snopeqop.d 𝐷 ∈ V
propeqop.e 𝐸 ∈ V
propeqop.f 𝐹 ∈ V
Assertion
Ref Expression
propeqop ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))

Proof of Theorem propeqop
StepHypRef Expression
1 snopeqop.a . . . . 5 𝐴 ∈ V
2 snopeqop.b . . . . 5 𝐵 ∈ V
3 propeqop.e . . . . 5 𝐸 ∈ V
41, 2, 3opeqsn 4932 . . . 4 (⟨𝐴, 𝐵⟩ = {𝐸} ↔ (𝐴 = 𝐵𝐸 = {𝐴}))
5 snopeqop.c . . . . 5 𝐶 ∈ V
6 snopeqop.d . . . . 5 𝐷 ∈ V
7 propeqop.f . . . . 5 𝐹 ∈ V
85, 6, 3, 7opeqpr 4933 . . . 4 (⟨𝐶, 𝐷⟩ = {𝐸, 𝐹} ↔ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})))
94, 8anbi12i 732 . . 3 ((⟨𝐴, 𝐵⟩ = {𝐸} ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) ↔ ((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))))
101, 2, 3, 7opeqpr 4933 . . . 4 (⟨𝐴, 𝐵⟩ = {𝐸, 𝐹} ↔ ((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})))
115, 6, 3opeqsn 4932 . . . 4 (⟨𝐶, 𝐷⟩ = {𝐸} ↔ (𝐶 = 𝐷𝐸 = {𝐶}))
1210, 11anbi12i 732 . . 3 ((⟨𝐴, 𝐵⟩ = {𝐸, 𝐹} ∧ ⟨𝐶, 𝐷⟩ = {𝐸}) ↔ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶})))
139, 12orbi12i 543 . 2 (((⟨𝐴, 𝐵⟩ = {𝐸} ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) ∨ (⟨𝐴, 𝐵⟩ = {𝐸, 𝐹} ∧ ⟨𝐶, 𝐷⟩ = {𝐸})) ↔ (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶}))))
14 eqcom 2633 . . 3 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ⟨𝐸, 𝐹⟩ = {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
15 opex 4898 . . . 4 𝐴, 𝐵⟩ ∈ V
16 opex 4898 . . . 4 𝐶, 𝐷⟩ ∈ V
173, 7, 15, 16opeqpr 4933 . . 3 (⟨𝐸, 𝐹⟩ = {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ ((⟨𝐴, 𝐵⟩ = {𝐸} ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) ∨ (⟨𝐴, 𝐵⟩ = {𝐸, 𝐹} ∧ ⟨𝐶, 𝐷⟩ = {𝐸})))
1814, 17bitri 264 . 2 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((⟨𝐴, 𝐵⟩ = {𝐸} ∧ ⟨𝐶, 𝐷⟩ = {𝐸, 𝐹}) ∨ (⟨𝐴, 𝐵⟩ = {𝐸, 𝐹} ∧ ⟨𝐶, 𝐷⟩ = {𝐸})))
19 simpl 473 . . . . . . . . 9 ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) → 𝐴 = 𝐵)
20 simpr 477 . . . . . . . . 9 ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐸 = {𝐴})
2119, 20anim12i 589 . . . . . . . 8 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (𝐴 = 𝐵𝐸 = {𝐴}))
22 sneq 4163 . . . . . . . . . . . . 13 (𝐴 = 𝐶 → {𝐴} = {𝐶})
2322eqeq2d 2636 . . . . . . . . . . . 12 (𝐴 = 𝐶 → (𝐸 = {𝐴} ↔ 𝐸 = {𝐶}))
2423biimpa 501 . . . . . . . . . . 11 ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐸 = {𝐶})
2524adantl 482 . . . . . . . . . 10 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → 𝐸 = {𝐶})
26 preq1 4243 . . . . . . . . . . . . . . 15 (𝐴 = 𝐶 → {𝐴, 𝐷} = {𝐶, 𝐷})
2726adantr 481 . . . . . . . . . . . . . 14 ((𝐴 = 𝐶𝐸 = {𝐴}) → {𝐴, 𝐷} = {𝐶, 𝐷})
2827eqeq2d 2636 . . . . . . . . . . . . 13 ((𝐴 = 𝐶𝐸 = {𝐴}) → (𝐹 = {𝐴, 𝐷} ↔ 𝐹 = {𝐶, 𝐷}))
2928biimpcd 239 . . . . . . . . . . . 12 (𝐹 = {𝐴, 𝐷} → ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐹 = {𝐶, 𝐷}))
3029adantl 482 . . . . . . . . . . 11 ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) → ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐹 = {𝐶, 𝐷}))
3130imp 445 . . . . . . . . . 10 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → 𝐹 = {𝐶, 𝐷})
3225, 31jca 554 . . . . . . . . 9 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}))
3332orcd 407 . . . . . . . 8 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})))
3421, 33jca 554 . . . . . . 7 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → ((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))))
3534orcd 407 . . . . . 6 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶}))))
3635ex 450 . . . . 5 ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) → ((𝐴 = 𝐶𝐸 = {𝐴}) → (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶})))))
37 simpr 477 . . . . . . . . . . 11 ((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) → 𝐹 = {𝐴, 𝐵})
3820, 37anim12i 589 . . . . . . . . . 10 (((𝐴 = 𝐶𝐸 = {𝐴}) ∧ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})) → (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}))
3938ancoms 469 . . . . . . . . 9 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}))
4039orcd 407 . . . . . . . 8 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → ((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})))
41 simpl 473 . . . . . . . . . . . . 13 ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐴 = 𝐶)
4241eqeq1d 2628 . . . . . . . . . . . 12 ((𝐴 = 𝐶𝐸 = {𝐴}) → (𝐴 = 𝐷𝐶 = 𝐷))
4342biimpcd 239 . . . . . . . . . . 11 (𝐴 = 𝐷 → ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐶 = 𝐷))
4443adantr 481 . . . . . . . . . 10 ((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) → ((𝐴 = 𝐶𝐸 = {𝐴}) → 𝐶 = 𝐷))
4544imp 445 . . . . . . . . 9 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → 𝐶 = 𝐷)
4623biimpd 219 . . . . . . . . . . . 12 (𝐴 = 𝐶 → (𝐸 = {𝐴} → 𝐸 = {𝐶}))
4746a1dd 50 . . . . . . . . . . 11 (𝐴 = 𝐶 → (𝐸 = {𝐴} → ((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) → 𝐸 = {𝐶})))
4847imp 445 . . . . . . . . . 10 ((𝐴 = 𝐶𝐸 = {𝐴}) → ((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) → 𝐸 = {𝐶}))
4948impcom 446 . . . . . . . . 9 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → 𝐸 = {𝐶})
5045, 49jca 554 . . . . . . . 8 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (𝐶 = 𝐷𝐸 = {𝐶}))
5140, 50jca 554 . . . . . . 7 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶})))
5251olcd 408 . . . . . 6 (((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) ∧ (𝐴 = 𝐶𝐸 = {𝐴})) → (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶}))))
5352ex 450 . . . . 5 ((𝐴 = 𝐷𝐹 = {𝐴, 𝐵}) → ((𝐴 = 𝐶𝐸 = {𝐴}) → (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶})))))
5436, 53jaoi 394 . . . 4 (((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})) → ((𝐴 = 𝐶𝐸 = {𝐴}) → (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶})))))
5554impcom 446 . . 3 (((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))) → (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶}))))
56 eqeq1 2630 . . . . . . . . . . . . 13 (𝐸 = {𝐶} → (𝐸 = {𝐴} ↔ {𝐶} = {𝐴}))
575sneqr 4344 . . . . . . . . . . . . . 14 ({𝐶} = {𝐴} → 𝐶 = 𝐴)
5857eqcomd 2632 . . . . . . . . . . . . 13 ({𝐶} = {𝐴} → 𝐴 = 𝐶)
5956, 58syl6bi 243 . . . . . . . . . . . 12 (𝐸 = {𝐶} → (𝐸 = {𝐴} → 𝐴 = 𝐶))
6059adantr 481 . . . . . . . . . . 11 ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) → (𝐸 = {𝐴} → 𝐴 = 𝐶))
61 eqeq1 2630 . . . . . . . . . . . . 13 (𝐸 = {𝐶, 𝐷} → (𝐸 = {𝐴} ↔ {𝐶, 𝐷} = {𝐴}))
625, 6, 1preqsn 4366 . . . . . . . . . . . . . 14 ({𝐶, 𝐷} = {𝐴} ↔ (𝐶 = 𝐷𝐷 = 𝐴))
63 eqtr 2645 . . . . . . . . . . . . . . 15 ((𝐶 = 𝐷𝐷 = 𝐴) → 𝐶 = 𝐴)
6463eqcomd 2632 . . . . . . . . . . . . . 14 ((𝐶 = 𝐷𝐷 = 𝐴) → 𝐴 = 𝐶)
6562, 64sylbi 207 . . . . . . . . . . . . 13 ({𝐶, 𝐷} = {𝐴} → 𝐴 = 𝐶)
6661, 65syl6bi 243 . . . . . . . . . . . 12 (𝐸 = {𝐶, 𝐷} → (𝐸 = {𝐴} → 𝐴 = 𝐶))
6766adantr 481 . . . . . . . . . . 11 ((𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}) → (𝐸 = {𝐴} → 𝐴 = 𝐶))
6860, 67jaoi 394 . . . . . . . . . 10 (((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) → (𝐸 = {𝐴} → 𝐴 = 𝐶))
6968com12 32 . . . . . . . . 9 (𝐸 = {𝐴} → (((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) → 𝐴 = 𝐶))
7069adantl 482 . . . . . . . 8 ((𝐴 = 𝐵𝐸 = {𝐴}) → (((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) → 𝐴 = 𝐶))
7170impcom 446 . . . . . . 7 ((((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → 𝐴 = 𝐶)
72 simpr 477 . . . . . . . 8 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐸 = {𝐴})
7372adantl 482 . . . . . . 7 ((((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → 𝐸 = {𝐴})
7471, 73jca 554 . . . . . 6 ((((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → (𝐴 = 𝐶𝐸 = {𝐴}))
75 simpl 473 . . . . . . . . . . 11 ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐴 = 𝐵)
7675adantr 481 . . . . . . . . . 10 (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ (𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷})) → 𝐴 = 𝐵)
77 eqeq1 2630 . . . . . . . . . . . . . . . . . 18 (𝐸 = {𝐴} → (𝐸 = {𝐶} ↔ {𝐴} = {𝐶}))
781sneqr 4344 . . . . . . . . . . . . . . . . . . 19 ({𝐴} = {𝐶} → 𝐴 = 𝐶)
7978eqcomd 2632 . . . . . . . . . . . . . . . . . 18 ({𝐴} = {𝐶} → 𝐶 = 𝐴)
8077, 79syl6bi 243 . . . . . . . . . . . . . . . . 17 (𝐸 = {𝐴} → (𝐸 = {𝐶} → 𝐶 = 𝐴))
8180adantl 482 . . . . . . . . . . . . . . . 16 ((𝐴 = 𝐵𝐸 = {𝐴}) → (𝐸 = {𝐶} → 𝐶 = 𝐴))
8281impcom 446 . . . . . . . . . . . . . . 15 ((𝐸 = {𝐶} ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → 𝐶 = 𝐴)
8382preq1d 4249 . . . . . . . . . . . . . 14 ((𝐸 = {𝐶} ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → {𝐶, 𝐷} = {𝐴, 𝐷})
8483eqeq2d 2636 . . . . . . . . . . . . 13 ((𝐸 = {𝐶} ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → (𝐹 = {𝐶, 𝐷} ↔ 𝐹 = {𝐴, 𝐷}))
8584biimpd 219 . . . . . . . . . . . 12 ((𝐸 = {𝐶} ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → (𝐹 = {𝐶, 𝐷} → 𝐹 = {𝐴, 𝐷}))
8685impancom 456 . . . . . . . . . . 11 ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) → ((𝐴 = 𝐵𝐸 = {𝐴}) → 𝐹 = {𝐴, 𝐷}))
8786impcom 446 . . . . . . . . . 10 (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ (𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷})) → 𝐹 = {𝐴, 𝐷})
8876, 87jca 554 . . . . . . . . 9 (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ (𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷})) → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))
8988ex 450 . . . . . . . 8 ((𝐴 = 𝐵𝐸 = {𝐴}) → ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷})))
90 eqcom 2633 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 = 𝐴𝐴 = 𝐷)
9190biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 = 𝐴𝐴 = 𝐷)
9291adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 = 𝐷𝐷 = 𝐴) → 𝐴 = 𝐷)
9392adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐷)
9493adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) ∧ 𝐹 = {𝐶}) → 𝐴 = 𝐷)
95 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
9664eqeq1d 2628 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶 = 𝐷𝐷 = 𝐴) → (𝐴 = 𝐵𝐶 = 𝐵))
9796biimpa 501 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → 𝐶 = 𝐵)
9897eqcomd 2632 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → 𝐵 = 𝐶)
991, 2, 5preqsn 4366 . . . . . . . . . . . . . . . . . . . . . 22 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
10095, 98, 99sylanbrc 697 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → {𝐴, 𝐵} = {𝐶})
101100eqcomd 2632 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → {𝐶} = {𝐴, 𝐵})
102101eqeq2d 2636 . . . . . . . . . . . . . . . . . . 19 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → (𝐹 = {𝐶} ↔ 𝐹 = {𝐴, 𝐵}))
103102biimpa 501 . . . . . . . . . . . . . . . . . 18 ((((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) ∧ 𝐹 = {𝐶}) → 𝐹 = {𝐴, 𝐵})
10494, 103jca 554 . . . . . . . . . . . . . . . . 17 ((((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) ∧ 𝐹 = {𝐶}) → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))
105104ex 450 . . . . . . . . . . . . . . . 16 (((𝐶 = 𝐷𝐷 = 𝐴) ∧ 𝐴 = 𝐵) → (𝐹 = {𝐶} → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))
106105ex 450 . . . . . . . . . . . . . . 15 ((𝐶 = 𝐷𝐷 = 𝐴) → (𝐴 = 𝐵 → (𝐹 = {𝐶} → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
107106com23 86 . . . . . . . . . . . . . 14 ((𝐶 = 𝐷𝐷 = 𝐴) → (𝐹 = {𝐶} → (𝐴 = 𝐵 → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
10862, 107sylbi 207 . . . . . . . . . . . . 13 ({𝐶, 𝐷} = {𝐴} → (𝐹 = {𝐶} → (𝐴 = 𝐵 → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
10961, 108syl6bi 243 . . . . . . . . . . . 12 (𝐸 = {𝐶, 𝐷} → (𝐸 = {𝐴} → (𝐹 = {𝐶} → (𝐴 = 𝐵 → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))))
110109com23 86 . . . . . . . . . . 11 (𝐸 = {𝐶, 𝐷} → (𝐹 = {𝐶} → (𝐸 = {𝐴} → (𝐴 = 𝐵 → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))))
111110imp 445 . . . . . . . . . 10 ((𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}) → (𝐸 = {𝐴} → (𝐴 = 𝐵 → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
112111com13 88 . . . . . . . . 9 (𝐴 = 𝐵 → (𝐸 = {𝐴} → ((𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}) → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
113112imp 445 . . . . . . . 8 ((𝐴 = 𝐵𝐸 = {𝐴}) → ((𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}) → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))
11489, 113orim12d 882 . . . . . . 7 ((𝐴 = 𝐵𝐸 = {𝐴}) → (((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) → ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
115114impcom 446 . . . . . 6 ((((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))
11674, 115jca 554 . . . . 5 ((((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶})) ∧ (𝐴 = 𝐵𝐸 = {𝐴})) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
117116ancoms 469 . . . 4 (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
118 eqeq1 2630 . . . . . . . . . . . . . . . 16 (𝐸 = {𝐶} → (𝐸 = {𝐴, 𝐵} ↔ {𝐶} = {𝐴, 𝐵}))
119 eqcom 2633 . . . . . . . . . . . . . . . . . 18 ({𝐶} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐶})
120119, 99bitri 264 . . . . . . . . . . . . . . . . 17 ({𝐶} = {𝐴, 𝐵} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
121 eqtr 2645 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
122121adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐸 = {𝐶}) → 𝐴 = 𝐶)
123121eqcomd 2632 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐶 = 𝐴)
124 sneq 4163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐶 = 𝐴 → {𝐶} = {𝐴})
125123, 124syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 = 𝐵𝐵 = 𝐶) → {𝐶} = {𝐴})
126125eqeq2d 2636 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐸 = {𝐶} ↔ 𝐸 = {𝐴}))
127126biimpa 501 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐸 = {𝐶}) → 𝐸 = {𝐴})
128122, 127jca 554 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐸 = {𝐶}) → (𝐴 = 𝐶𝐸 = {𝐴}))
129128ex 450 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐸 = {𝐶} → (𝐴 = 𝐶𝐸 = {𝐴})))
130129a1d 25 . . . . . . . . . . . . . . . . . . 19 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐹 = {𝐴} → (𝐸 = {𝐶} → (𝐴 = 𝐶𝐸 = {𝐴}))))
131130a1i 11 . . . . . . . . . . . . . . . . . 18 (𝐶 = 𝐷 → ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐹 = {𝐴} → (𝐸 = {𝐶} → (𝐴 = 𝐶𝐸 = {𝐴})))))
132131com14 96 . . . . . . . . . . . . . . . . 17 (𝐸 = {𝐶} → ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐹 = {𝐴} → (𝐶 = 𝐷 → (𝐴 = 𝐶𝐸 = {𝐴})))))
133120, 132syl5bi 232 . . . . . . . . . . . . . . . 16 (𝐸 = {𝐶} → ({𝐶} = {𝐴, 𝐵} → (𝐹 = {𝐴} → (𝐶 = 𝐷 → (𝐴 = 𝐶𝐸 = {𝐴})))))
134118, 133sylbid 230 . . . . . . . . . . . . . . 15 (𝐸 = {𝐶} → (𝐸 = {𝐴, 𝐵} → (𝐹 = {𝐴} → (𝐶 = 𝐷 → (𝐴 = 𝐶𝐸 = {𝐴})))))
135134com24 95 . . . . . . . . . . . . . 14 (𝐸 = {𝐶} → (𝐶 = 𝐷 → (𝐹 = {𝐴} → (𝐸 = {𝐴, 𝐵} → (𝐴 = 𝐶𝐸 = {𝐴})))))
136135impcom 446 . . . . . . . . . . . . 13 ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐹 = {𝐴} → (𝐸 = {𝐴, 𝐵} → (𝐴 = 𝐶𝐸 = {𝐴}))))
137136com13 88 . . . . . . . . . . . 12 (𝐸 = {𝐴, 𝐵} → (𝐹 = {𝐴} → ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐴 = 𝐶𝐸 = {𝐴}))))
138137imp 445 . . . . . . . . . . 11 ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) → ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐴 = 𝐶𝐸 = {𝐴})))
13959adantl 482 . . . . . . . . . . . . . . . 16 ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐸 = {𝐴} → 𝐴 = 𝐶))
140139com12 32 . . . . . . . . . . . . . . 15 (𝐸 = {𝐴} → ((𝐶 = 𝐷𝐸 = {𝐶}) → 𝐴 = 𝐶))
141140adantr 481 . . . . . . . . . . . . . 14 ((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) → ((𝐶 = 𝐷𝐸 = {𝐶}) → 𝐴 = 𝐶))
142141imp 445 . . . . . . . . . . . . 13 (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∧ (𝐶 = 𝐷𝐸 = {𝐶})) → 𝐴 = 𝐶)
143 simpl 473 . . . . . . . . . . . . . 14 ((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) → 𝐸 = {𝐴})
144143adantr 481 . . . . . . . . . . . . 13 (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∧ (𝐶 = 𝐷𝐸 = {𝐶})) → 𝐸 = {𝐴})
145142, 144jca 554 . . . . . . . . . . . 12 (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∧ (𝐶 = 𝐷𝐸 = {𝐶})) → (𝐴 = 𝐶𝐸 = {𝐴}))
146145ex 450 . . . . . . . . . . 11 ((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) → ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐴 = 𝐶𝐸 = {𝐴})))
147138, 146jaoi 394 . . . . . . . . . 10 (((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵})) → ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐴 = 𝐶𝐸 = {𝐴})))
148147impcom 446 . . . . . . . . 9 (((𝐶 = 𝐷𝐸 = {𝐶}) ∧ ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}))) → (𝐴 = 𝐶𝐸 = {𝐴}))
149 eqeq1 2630 . . . . . . . . . . . . . . . 16 (𝐸 = {𝐴, 𝐵} → (𝐸 = {𝐶} ↔ {𝐴, 𝐵} = {𝐶}))
150 simpl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐵)
151150adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) → 𝐴 = 𝐵)
152151adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) ∧ 𝐹 = {𝐴}) → 𝐴 = 𝐵)
153 eqtr 2645 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 = 𝐶𝐶 = 𝐷) → 𝐴 = 𝐷)
154 dfsn2 4166 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝐴} = {𝐴, 𝐴}
155 preq2 4244 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐴 = 𝐷 → {𝐴, 𝐴} = {𝐴, 𝐷})
156154, 155syl5eq 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴 = 𝐷 → {𝐴} = {𝐴, 𝐷})
157153, 156syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 = 𝐶𝐶 = 𝐷) → {𝐴} = {𝐴, 𝐷})
158157ex 450 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐶 → (𝐶 = 𝐷 → {𝐴} = {𝐴, 𝐷}))
159121, 158syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐶 = 𝐷 → {𝐴} = {𝐴, 𝐷}))
160159imp 445 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) → {𝐴} = {𝐴, 𝐷})
161160eqeq2d 2636 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) → (𝐹 = {𝐴} ↔ 𝐹 = {𝐴, 𝐷}))
162161biimpa 501 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) ∧ 𝐹 = {𝐴}) → 𝐹 = {𝐴, 𝐷})
163152, 162jca 554 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) ∧ 𝐹 = {𝐴}) → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))
164163ex 450 . . . . . . . . . . . . . . . . . . 19 (((𝐴 = 𝐵𝐵 = 𝐶) ∧ 𝐶 = 𝐷) → (𝐹 = {𝐴} → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷})))
165164ex 450 . . . . . . . . . . . . . . . . . 18 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐶 = 𝐷 → (𝐹 = {𝐴} → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))))
166165com23 86 . . . . . . . . . . . . . . . . 17 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐹 = {𝐴} → (𝐶 = 𝐷 → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))))
16799, 166sylbi 207 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} = {𝐶} → (𝐹 = {𝐴} → (𝐶 = 𝐷 → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))))
168149, 167syl6bi 243 . . . . . . . . . . . . . . 15 (𝐸 = {𝐴, 𝐵} → (𝐸 = {𝐶} → (𝐹 = {𝐴} → (𝐶 = 𝐷 → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷})))))
169168com23 86 . . . . . . . . . . . . . 14 (𝐸 = {𝐴, 𝐵} → (𝐹 = {𝐴} → (𝐸 = {𝐶} → (𝐶 = 𝐷 → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷})))))
170169imp 445 . . . . . . . . . . . . 13 ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) → (𝐸 = {𝐶} → (𝐶 = 𝐷 → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))))
171170com13 88 . . . . . . . . . . . 12 (𝐶 = 𝐷 → (𝐸 = {𝐶} → ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷}))))
172171imp 445 . . . . . . . . . . 11 ((𝐶 = 𝐷𝐸 = {𝐶}) → ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) → (𝐴 = 𝐵𝐹 = {𝐴, 𝐷})))
17380imp 445 . . . . . . . . . . . . . . . . 17 ((𝐸 = {𝐴} ∧ 𝐸 = {𝐶}) → 𝐶 = 𝐴)
174173eqeq1d 2628 . . . . . . . . . . . . . . . 16 ((𝐸 = {𝐴} ∧ 𝐸 = {𝐶}) → (𝐶 = 𝐷𝐴 = 𝐷))
175174biimpd 219 . . . . . . . . . . . . . . 15 ((𝐸 = {𝐴} ∧ 𝐸 = {𝐶}) → (𝐶 = 𝐷𝐴 = 𝐷))
176175ex 450 . . . . . . . . . . . . . 14 (𝐸 = {𝐴} → (𝐸 = {𝐶} → (𝐶 = 𝐷𝐴 = 𝐷)))
177176com13 88 . . . . . . . . . . . . 13 (𝐶 = 𝐷 → (𝐸 = {𝐶} → (𝐸 = {𝐴} → 𝐴 = 𝐷)))
178177imp 445 . . . . . . . . . . . 12 ((𝐶 = 𝐷𝐸 = {𝐶}) → (𝐸 = {𝐴} → 𝐴 = 𝐷))
179178anim1d 587 . . . . . . . . . . 11 ((𝐶 = 𝐷𝐸 = {𝐶}) → ((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) → (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))
180172, 179orim12d 882 . . . . . . . . . 10 ((𝐶 = 𝐷𝐸 = {𝐶}) → (((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵})) → ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
181180imp 445 . . . . . . . . 9 (((𝐶 = 𝐷𝐸 = {𝐶}) ∧ ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}))) → ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))
182148, 181jca 554 . . . . . . . 8 (((𝐶 = 𝐷𝐸 = {𝐶}) ∧ ((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}))) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
183182ex 450 . . . . . . 7 ((𝐶 = 𝐷𝐸 = {𝐶}) → (((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵})) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))))
184183com12 32 . . . . . 6 (((𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴}) ∨ (𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵})) → ((𝐶 = 𝐷𝐸 = {𝐶}) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))))
185184orcoms 404 . . . . 5 (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) → ((𝐶 = 𝐷𝐸 = {𝐶}) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵})))))
186185imp 445 . . . 4 ((((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶})) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
187117, 186jaoi 394 . . 3 ((((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶}))) → ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
18855, 187impbii 199 . 2 (((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))) ↔ (((𝐴 = 𝐵𝐸 = {𝐴}) ∧ ((𝐸 = {𝐶} ∧ 𝐹 = {𝐶, 𝐷}) ∨ (𝐸 = {𝐶, 𝐷} ∧ 𝐹 = {𝐶}))) ∨ (((𝐸 = {𝐴} ∧ 𝐹 = {𝐴, 𝐵}) ∨ (𝐸 = {𝐴, 𝐵} ∧ 𝐹 = {𝐴})) ∧ (𝐶 = 𝐷𝐸 = {𝐶}))))
18913, 18, 1883bitr4i 292 1 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  {cpr 4155  cop 4159
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160
This theorem is referenced by:  propssopi  4936  fun2dmnopgexmpl  40588
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