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Theorem eqerOLD 7665
 Description: Obsolete proof of eqer 7664 as of 1-May-2021. Equivalence relation involving equality of dependent classes 𝐴(𝑥) and 𝐵(𝑦). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eqer.1 (𝑥 = 𝑦𝐴 = 𝐵)
eqer.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
Assertion
Ref Expression
eqerOLD 𝑅 Er V
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem eqerOLD
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.2 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
21relopabi 5167 . . . 4 Rel 𝑅
32a1i 11 . . 3 (⊤ → Rel 𝑅)
4 id 22 . . . . . 6 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
54eqcomd 2616 . . . . 5 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
6 eqer.1 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝐵)
76, 1eqerlem 7663 . . . . 5 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
86, 1eqerlem 7663 . . . . 5 (𝑤𝑅𝑧𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
95, 7, 83imtr4i 280 . . . 4 (𝑧𝑅𝑤𝑤𝑅𝑧)
109adantl 481 . . 3 ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧)
11 eqtr 2629 . . . . 5 ((𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
126, 1eqerlem 7663 . . . . . 6 (𝑤𝑅𝑣𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
137, 12anbi12i 729 . . . . 5 ((𝑧𝑅𝑤𝑤𝑅𝑣) ↔ (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴))
146, 1eqerlem 7663 . . . . 5 (𝑧𝑅𝑣𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
1511, 13, 143imtr4i 280 . . . 4 ((𝑧𝑅𝑤𝑤𝑅𝑣) → 𝑧𝑅𝑣)
1615adantl 481 . . 3 ((⊤ ∧ (𝑧𝑅𝑤𝑤𝑅𝑣)) → 𝑧𝑅𝑣)
17 vex 3176 . . . . 5 𝑧 ∈ V
18 eqid 2610 . . . . . 6 𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
196, 1eqerlem 7663 . . . . . 6 (𝑧𝑅𝑧𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2018, 19mpbir 220 . . . . 5 𝑧𝑅𝑧
2117, 202th 253 . . . 4 (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
2221a1i 11 . . 3 (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧))
233, 10, 16, 22iserd 7655 . 2 (⊤ → 𝑅 Er V)
2423trud 1484 1 𝑅 Er V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499   class class class wbr 4583  {copab 4642  Rel wrel 5043   Er wer 7626 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-er 7629 This theorem is referenced by: (None)
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