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Theorem eqer 8063
Description: 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 shortened by AV, 1-May-2021.)
Hypotheses
Ref Expression
eqer.1 (𝑥 = 𝑦𝐴 = 𝐵)
eqer.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
Assertion
Ref Expression
eqer 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 eqer
Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.2 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
21relopabi 5493 . 2 Rel 𝑅
3 id 22 . . . 4 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
43eqcomd 2784 . . 3 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
5 eqer.1 . . . 4 (𝑥 = 𝑦𝐴 = 𝐵)
65, 1eqerlem 8062 . . 3 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
75, 1eqerlem 8062 . . 3 (𝑤𝑅𝑧𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
84, 6, 73imtr4i 284 . 2 (𝑧𝑅𝑤𝑤𝑅𝑧)
9 eqtr 2799 . . 3 ((𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
105, 1eqerlem 8062 . . . 4 (𝑤𝑅𝑣𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
116, 10anbi12i 620 . . 3 ((𝑧𝑅𝑤𝑤𝑅𝑣) ↔ (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴))
125, 1eqerlem 8062 . . 3 (𝑧𝑅𝑣𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
139, 11, 123imtr4i 284 . 2 ((𝑧𝑅𝑤𝑤𝑅𝑣) → 𝑧𝑅𝑣)
14 vex 3401 . . 3 𝑧 ∈ V
15 eqid 2778 . . . 4 𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
165, 1eqerlem 8062 . . . 4 (𝑧𝑅𝑧𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
1715, 16mpbir 223 . . 3 𝑧𝑅𝑧
1814, 172th 256 . 2 (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
192, 8, 13, 18iseri 8055 1 𝑅 Er V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  csb 3751   class class class wbr 4888  {copab 4950   Er wer 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-er 8028
This theorem is referenced by:  ider  8064  frgpuplem  18582  fneer  32944
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