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Theorem eqer 6074
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 4406 . . . 4 Rel 𝑅
32a1i 9 . . 3 ( ⊤ → Rel 𝑅)
4 id 19 . . . . . 6 (z / xA = w / xAz / xA = w / xA)
54eqcomd 2042 . . . . 5 (z / xA = w / xAw / xA = z / xA)
6 eqer.1 . . . . . 6 (x = yA = B)
76, 1eqerlem 6073 . . . . 5 (z𝑅wz / xA = w / xA)
86, 1eqerlem 6073 . . . . 5 (w𝑅zw / xA = z / xA)
95, 7, 83imtr4i 190 . . . 4 (z𝑅ww𝑅z)
109adantl 262 . . 3 (( ⊤ z𝑅w) → w𝑅z)
11 eqtr 2054 . . . . 5 ((z / xA = w / xA w / xA = v / xA) → z / xA = v / xA)
126, 1eqerlem 6073 . . . . . 6 (w𝑅vw / xA = v / xA)
137, 12anbi12i 433 . . . . 5 ((z𝑅w w𝑅v) ↔ (z / xA = w / xA w / xA = v / xA))
146, 1eqerlem 6073 . . . . 5 (z𝑅vz / xA = v / xA)
1511, 13, 143imtr4i 190 . . . 4 ((z𝑅w w𝑅v) → z𝑅v)
1615adantl 262 . . 3 (( ⊤ (z𝑅w w𝑅v)) → z𝑅v)
17 vex 2554 . . . . 5 z V
18 eqid 2037 . . . . . 6 z / xA = z / xA
196, 1eqerlem 6073 . . . . . 6 (z𝑅zz / xA = z / xA)
2018, 19mpbir 134 . . . . 5 z𝑅z
2117, 202th 163 . . . 4 (z V ↔ z𝑅z)
2221a1i 9 . . 3 ( ⊤ → (z V ↔ z𝑅z))
233, 10, 16, 22iserd 6068 . 2 ( ⊤ → 𝑅 Er V)
2423trud 1251 1 𝑅 Er V
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wtru 1243   wcel 1390  Vcvv 2551  csb 2846   class class class wbr 3755  {copab 3808  Rel wrel 4293   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by:  ider  6075
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