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Theorem eqer 6168
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.)
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 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
21relopabi 4490 . . . 4 Rel 𝑅
32a1i 9 . . 3 (⊤ → Rel 𝑅)
4 id 19 . . . . . 6 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
54eqcomd 2061 . . . . 5 (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
6 eqer.1 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝐵)
76, 1eqerlem 6167 . . . . 5 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
86, 1eqerlem 6167 . . . . 5 (𝑤𝑅𝑧𝑤 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
95, 7, 83imtr4i 194 . . . 4 (𝑧𝑅𝑤𝑤𝑅𝑧)
109adantl 266 . . 3 ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧)
11 eqtr 2073 . . . . 5 ((𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
126, 1eqerlem 6167 . . . . . 6 (𝑤𝑅𝑣𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
137, 12anbi12i 441 . . . . 5 ((𝑧𝑅𝑤𝑤𝑅𝑣) ↔ (𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴𝑤 / 𝑥𝐴 = 𝑣 / 𝑥𝐴))
146, 1eqerlem 6167 . . . . 5 (𝑧𝑅𝑣𝑧 / 𝑥𝐴 = 𝑣 / 𝑥𝐴)
1511, 13, 143imtr4i 194 . . . 4 ((𝑧𝑅𝑤𝑤𝑅𝑣) → 𝑧𝑅𝑣)
1615adantl 266 . . 3 ((⊤ ∧ (𝑧𝑅𝑤𝑤𝑅𝑣)) → 𝑧𝑅𝑣)
17 vex 2577 . . . . 5 𝑧 ∈ V
18 eqid 2056 . . . . . 6 𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
196, 1eqerlem 6167 . . . . . 6 (𝑧𝑅𝑧𝑧 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2018, 19mpbir 138 . . . . 5 𝑧𝑅𝑧
2117, 202th 167 . . . 4 (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
2221a1i 9 . . 3 (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧))
233, 10, 16, 22iserd 6162 . 2 (⊤ → 𝑅 Er V)
2423trud 1268 1 𝑅 Er V
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wtru 1260  wcel 1409  Vcvv 2574  csb 2879   class class class wbr 3791  {copab 3844  Rel wrel 4377   Er wer 6133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-er 6136
This theorem is referenced by:  ider  6169
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