Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq1b Structured version   Visualization version   GIF version

Theorem preq1b 4317
 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 4239 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
31, 2syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴, 𝐶})
4 eleq2 2677 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
53, 4syl5ibcom 234 . . . . . 6 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
6 elprg 4144 . . . . . . 7 (𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
71, 6syl 17 . . . . . 6 (𝜑 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
85, 7sylibd 228 . . . . 5 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶)))
98imp 444 . . . 4 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵𝐴 = 𝐶))
10 preq1b.b . . . . . . . 8 (𝜑𝐵𝑊)
11 prid1g 4239 . . . . . . . 8 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
1210, 11syl 17 . . . . . . 7 (𝜑𝐵 ∈ {𝐵, 𝐶})
13 eleq2 2677 . . . . . . 7 ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
1412, 13syl5ibrcom 236 . . . . . 6 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
15 elprg 4144 . . . . . . 7 (𝐵𝑊 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1610, 15syl 17 . . . . . 6 (𝜑 → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐶)))
1714, 16sylibd 228 . . . . 5 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴𝐵 = 𝐶)))
1817imp 444 . . . 4 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴𝐵 = 𝐶))
19 eqcom 2617 . . . 4 (𝐴 = 𝐵𝐵 = 𝐴)
20 eqeq2 2621 . . . 4 (𝐴 = 𝐶 → (𝐵 = 𝐴𝐵 = 𝐶))
219, 18, 19, 20oplem1 999 . . 3 ((𝜑 ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
2221ex 449 . 2 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
23 preq1 4212 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2422, 23impbid1 214 1 (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  preq2b  4318  preqr1  4319  preqr1g  4325  uhgr3cyclexlem  41348
 Copyright terms: Public domain W3C validator