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Theorem eqer 6049
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 4390 . . . 4 Rel 𝑅
32a1i 9 . . 3 ( ⊤ → Rel 𝑅)
4 id 19 . . . . . 6 (z / xA = w / xAz / xA = w / xA)
54eqcomd 2027 . . . . 5 (z / xA = w / xAw / xA = z / xA)
6 eqer.1 . . . . . 6 (x = yA = B)
76, 1eqerlem 6048 . . . . 5 (z𝑅wz / xA = w / xA)
86, 1eqerlem 6048 . . . . 5 (w𝑅zw / xA = z / xA)
95, 7, 83imtr4i 190 . . . 4 (z𝑅ww𝑅z)
109adantl 262 . . 3 (( ⊤ z𝑅w) → w𝑅z)
11 eqtr 2039 . . . . 5 ((z / xA = w / xA w / xA = v / xA) → z / xA = v / xA)
126, 1eqerlem 6048 . . . . . 6 (w𝑅vw / xA = v / xA)
137, 12anbi12i 436 . . . . 5 ((z𝑅w w𝑅v) ↔ (z / xA = w / xA w / xA = v / xA))
146, 1eqerlem 6048 . . . . 5 (z𝑅vz / xA = v / xA)
1511, 13, 143imtr4i 190 . . . 4 ((z𝑅w w𝑅v) → z𝑅v)
1615adantl 262 . . 3 (( ⊤ (z𝑅w w𝑅v)) → z𝑅v)
17 vex 2538 . . . . 5 z V
18 eqid 2022 . . . . . 6 z / xA = z / xA
196, 1eqerlem 6048 . . . . . 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 6043 . 2 ( ⊤ → 𝑅 Er V)
2423trud 1237 1 𝑅 Er V
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wtru 1229   wcel 1374  Vcvv 2535  csb 2829   class class class wbr 3738  {copab 3791  Rel wrel 4277   Er wer 6014
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-er 6017
This theorem is referenced by:  ider  6050
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