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Theorem or2expropbi 43794
 Description: If two classes are strictly ordered, there is an ordered pair of both classes fulfilling a wff iff there is an unordered pair of both classes fulfilling the wff. (Contributed by AV, 26-Aug-2023.)
Assertion
Ref Expression
or2expropbi (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑)) ↔ ∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑))))
Distinct variable groups:   𝑎,𝑏,𝐴   𝐵,𝑎,𝑏   𝑅,𝑎,𝑏   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)

Proof of Theorem or2expropbi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . 4 𝑎((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵))
2 nfv 1915 . . . . . . 7 𝑎𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
3 nfcv 2955 . . . . . . . 8 𝑎𝑦
4 nfsbc1v 3742 . . . . . . . 8 𝑎[𝑥 / 𝑎](𝑎𝑅𝑏𝜑)
53, 4nfsbcw 3744 . . . . . . 7 𝑎[𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)
62, 5nfan 1900 . . . . . 6 𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))
76nfex 2332 . . . . 5 𝑎𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))
87nfex 2332 . . . 4 𝑎𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))
9 nfv 1915 . . . . 5 𝑏((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵))
10 nfv 1915 . . . . . . . 8 𝑏𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
11 nfsbc1v 3742 . . . . . . . 8 𝑏[𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)
1210, 11nfan 1900 . . . . . . 7 𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))
1312nfex 2332 . . . . . 6 𝑏𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))
1413nfex 2332 . . . . 5 𝑏𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))
15 vex 3445 . . . . . . . . . 10 𝑎 ∈ V
16 vex 3445 . . . . . . . . . 10 𝑏 ∈ V
17 preq12bg 4747 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑋) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
1815, 16, 17mpanr12 704 . . . . . . . . 9 ((𝐴𝑋𝐵𝑋) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
19183adant3 1129 . . . . . . . 8 ((𝐴𝑋𝐵𝑋𝐴𝑅𝐵) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
2019adantl 485 . . . . . . 7 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
21 or2expropbilem1 43792 . . . . . . . . . 10 ((𝐴𝑋𝐵𝑋) → ((𝐴 = 𝑎𝐵 = 𝑏) → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
22213adant3 1129 . . . . . . . . 9 ((𝐴𝑋𝐵𝑋𝐴𝑅𝐵) → ((𝐴 = 𝑎𝐵 = 𝑏) → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
2322adantl 485 . . . . . . . 8 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → ((𝐴 = 𝑎𝐵 = 𝑏) → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
24 breq12 5039 . . . . . . . . . . . . 13 ((𝐵 = 𝑎𝐴 = 𝑏) → (𝐵𝑅𝐴𝑎𝑅𝑏))
2524ancoms 462 . . . . . . . . . . . 12 ((𝐴 = 𝑏𝐵 = 𝑎) → (𝐵𝑅𝐴𝑎𝑅𝑏))
2625adantl 485 . . . . . . . . . . 11 ((((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → (𝐵𝑅𝐴𝑎𝑅𝑏))
27 soasym 5472 . . . . . . . . . . . . . . . . 17 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝑅𝐵 → ¬ 𝐵𝑅𝐴))
2827ex 416 . . . . . . . . . . . . . . . 16 (𝑅 Or 𝑋 → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → ¬ 𝐵𝑅𝐴)))
2928adantl 485 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑅 Or 𝑋) → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 → ¬ 𝐵𝑅𝐴)))
3029expd 419 . . . . . . . . . . . . . 14 ((𝑋𝑉𝑅 Or 𝑋) → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝑅𝐵 → ¬ 𝐵𝑅𝐴))))
31303imp2 1346 . . . . . . . . . . . . 13 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → ¬ 𝐵𝑅𝐴)
3231pm2.21d 121 . . . . . . . . . . . 12 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (𝐵𝑅𝐴 → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
3332adantr 484 . . . . . . . . . . 11 ((((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → (𝐵𝑅𝐴 → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
3426, 33sylbird 263 . . . . . . . . . 10 ((((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → (𝑎𝑅𝑏 → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
3534impd 414 . . . . . . . . 9 ((((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))))
3635ex 416 . . . . . . . 8 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → ((𝐴 = 𝑏𝐵 = 𝑎) → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
3723, 36jaod 856 . . . . . . 7 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎)) → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
3820, 37sylbid 243 . . . . . 6 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → ({𝐴, 𝐵} = {𝑎, 𝑏} → ((𝑎𝑅𝑏𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))))
3938impd 414 . . . . 5 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))))
409, 14, 39exlimd 2216 . . . 4 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))))
411, 8, 40exlimd 2216 . . 3 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑))))
42 or2expropbilem2 43793 . . 3 (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑)) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎](𝑎𝑅𝑏𝜑)))
4341, 42syl6ibr 255 . 2 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑))))
44 oppr 43790 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → {𝐴, 𝐵} = {𝑎, 𝑏}))
4544anim1d 613 . . . . 5 ((𝐴𝑋𝐵𝑋) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑)) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑))))
46452eximdv 1920 . . . 4 ((𝐴𝑋𝐵𝑋) → (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑))))
47463adant3 1129 . . 3 ((𝐴𝑋𝐵𝑋𝐴𝑅𝐵) → (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑))))
4847adantl 485 . 2 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑)) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑))))
4943, 48impbid 215 1 (((𝑋𝑉𝑅 Or 𝑋) ∧ (𝐴𝑋𝐵𝑋𝐴𝑅𝐵)) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏𝜑)) ↔ ∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3442  [wsbc 3722  {cpr 4530  ⟨cop 4534   class class class wbr 5034   Or wor 5441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-po 5442  df-so 5443 This theorem is referenced by: (None)
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