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Theorem preq1b 4770
Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first elements are equal. (Contributed by AV, 18-Dec-2020.)
Hypotheses
Ref Expression
preq1b.a (𝜑𝐴𝑉)
preq1b.b (𝜑𝐵𝑊)
Assertion
Ref Expression
preq1b (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))

Proof of Theorem preq1b
StepHypRef Expression
1 preq1b.a . . . . . . . 8 (𝜑𝐴𝑉)
2 prid1g 4689 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
31, 2syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴, 𝐶})
4 eleq2 2901 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
53, 4syl5ibcom 247 . . . . . 6 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
6 elprg 4581 . . . . . . 7 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
71, 6syl 17 . . . . . 6 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
85, 7sylibd 241 . . . . 5 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
98imp 409 . . . 4 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵𝐴 = 𝐶))
10 preq1b.b . . . . . . . 8 (𝜑𝐵𝑊)
11 prid1g 4689 . . . . . . . 8 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
1210, 11syl 17 . . . . . . 7 (𝜑𝐵 ∈ {𝐵, 𝐶})
13 eleq2 2901 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
1412, 13syl5ibrcom 249 . . . . . 6 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
15 elprg 4581 . . . . . . 7 (𝐵𝑊 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1610, 15syl 17 . . . . . 6 (𝜑 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1714, 16sylibd 241 . . . . 5 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1817imp 409 . . . 4 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴𝐵 = 𝐶))
19 eqcom 2828 . . . 4 (𝐴 = 𝐵𝐵 = 𝐴)
20 eqeq2 2833 . . . 4 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
219, 18, 19, 20oplem1 1051 . . 3 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
2221ex 415 . 2 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
23 preq1 4662 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2422, 23impbid1 227 1 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1533  wcel 2110  {cpr 4562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-sn 4561  df-pr 4563
This theorem is referenced by:  preq2b  4771  preqr1  4772  preqr1g  4776  uhgr3cyclexlem  27954
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