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Theorem preq1b 4531
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 4452 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
31, 2syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴, 𝐶})
4 eleq2 2833 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
53, 4syl5ibcom 236 . . . . . 6 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
6 elprg 4357 . . . . . . 7 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
71, 6syl 17 . . . . . 6 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
85, 7sylibd 230 . . . . 5 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
98imp 395 . . . 4 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵𝐴 = 𝐶))
10 preq1b.b . . . . . . . 8 (𝜑𝐵𝑊)
11 prid1g 4452 . . . . . . . 8 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
1210, 11syl 17 . . . . . . 7 (𝜑𝐵 ∈ {𝐵, 𝐶})
13 eleq2 2833 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
1412, 13syl5ibrcom 238 . . . . . 6 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
15 elprg 4357 . . . . . . 7 (𝐵𝑊 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1610, 15syl 17 . . . . . 6 (𝜑 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1714, 16sylibd 230 . . . . 5 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1817imp 395 . . . 4 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴𝐵 = 𝐶))
19 eqcom 2772 . . . 4 (𝐴 = 𝐵𝐵 = 𝐴)
20 eqeq2 2776 . . . 4 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
219, 18, 19, 20oplem1 1079 . . 3 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
2221ex 401 . 2 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
23 preq1 4425 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2422, 23impbid1 216 1 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  {cpr 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3739  df-sn 4337  df-pr 4339
This theorem is referenced by:  preq2b  4532  preqr1  4533  preqr1g  4538  uhgr3cyclexlem  27477
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