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Theorem eqerOLD 7724
Description: Obsolete proof of eqer 7723 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 5210 . . . 4 Rel 𝑅
32a1i 11 . . 3 (⊤ → Rel 𝑅)
4 id 22 . . . . . 6 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
54eqcomd 2632 . . . . 5 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
6 eqer.1 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝐵)
76, 1eqerlem 7722 . . . . 5 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
86, 1eqerlem 7722 . . . . 5 (𝑤𝑅𝑧𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
95, 7, 83imtr4i 281 . . . 4 (𝑧𝑅𝑤𝑤𝑅𝑧)
109adantl 482 . . 3 ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧)
11 eqtr 2645 . . . . 5 ((𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
126, 1eqerlem 7722 . . . . . 6 (𝑤𝑅𝑣𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
137, 12anbi12i 732 . . . . 5 ((𝑧𝑅𝑤𝑤𝑅𝑣) ↔ (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴))
146, 1eqerlem 7722 . . . . 5 (𝑧𝑅𝑣𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
1511, 13, 143imtr4i 281 . . . 4 ((𝑧𝑅𝑤𝑤𝑅𝑣) → 𝑧𝑅𝑣)
1615adantl 482 . . 3 ((⊤ ∧ (𝑧𝑅𝑤𝑤𝑅𝑣)) → 𝑧𝑅𝑣)
17 vex 3194 . . . . 5 𝑧 ∈ V
18 eqid 2626 . . . . . 6 𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
196, 1eqerlem 7722 . . . . . 6 (𝑧𝑅𝑧𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2018, 19mpbir 221 . . . . 5 𝑧𝑅𝑧
2117, 202th 254 . . . 4 (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
2221a1i 11 . . 3 (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧))
233, 10, 16, 22iserd 7714 . 2 (⊤ → 𝑅 Er V)
2423trud 1490 1 𝑅 Er V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1992  Vcvv 3191  csb 3519   class class class wbr 4618  {copab 4677  Rel wrel 5084   Er wer 7685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-er 7688
This theorem is referenced by: (None)
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