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Theorem eqerlem 6048
Description: Lemma for eqer 6049. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
eqer.1 (x = yA = B)
eqer.2 𝑅 = {⟨x, y⟩ ∣ A = B}
Assertion
Ref Expression
eqerlem (z𝑅wz / xA = w / xA)
Distinct variable groups:   x,w,y   x,z,y   y,A   x,B
Allowed substitution hints:   A(x,z,w)   B(y,z,w)   𝑅(x,y,z,w)

Proof of Theorem eqerlem
StepHypRef Expression
1 eqer.2 . . 3 𝑅 = {⟨x, y⟩ ∣ A = B}
21brabsb 3972 . 2 (z𝑅w[z / x][w / y]A = B)
3 vex 2538 . . 3 z V
4 nfcsb1v 2859 . . . . 5 xz / xA
5 nfcsb1v 2859 . . . . 5 xw / xA
64, 5nfeq 2167 . . . 4 xz / xA = w / xA
7 vex 2538 . . . . . 6 w V
8 nfv 1402 . . . . . . 7 y A = w / xA
9 vex 2538 . . . . . . . . . 10 y V
10 nfcv 2160 . . . . . . . . . 10 xB
11 eqer.1 . . . . . . . . . 10 (x = yA = B)
129, 10, 11csbief 2868 . . . . . . . . 9 y / xA = B
13 csbeq1 2832 . . . . . . . . 9 (y = wy / xA = w / xA)
1412, 13syl5eqr 2068 . . . . . . . 8 (y = wB = w / xA)
1514eqeq2d 2033 . . . . . . 7 (y = w → (A = BA = w / xA))
168, 15sbciegf 2771 . . . . . 6 (w V → ([w / y]A = BA = w / xA))
177, 16ax-mp 7 . . . . 5 ([w / y]A = BA = w / xA)
18 csbeq1a 2837 . . . . . 6 (x = zA = z / xA)
1918eqeq1d 2030 . . . . 5 (x = z → (A = w / xAz / xA = w / xA))
2017, 19syl5bb 181 . . . 4 (x = z → ([w / y]A = Bz / xA = w / xA))
216, 20sbciegf 2771 . . 3 (z V → ([z / x][w / y]A = Bz / xA = w / xA))
223, 21ax-mp 7 . 2 ([z / x][w / y]A = Bz / xA = w / xA)
232, 22bitri 173 1 (z𝑅wz / xA = w / xA)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  [wsbc 2741  csb 2829   class class class wbr 3738  {copab 3791
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-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
This theorem is referenced by:  eqer  6049
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