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Theorem eqerlem 6073
 Description: Lemma for eqer 6074. (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 3989 . 2 (z𝑅w[z / x][w / y]A = B)
3 vex 2554 . . 3 z V
4 nfcsb1v 2876 . . . . 5 xz / xA
5 nfcsb1v 2876 . . . . 5 xw / xA
64, 5nfeq 2182 . . . 4 xz / xA = w / xA
7 vex 2554 . . . . . 6 w V
8 nfv 1418 . . . . . . 7 y A = w / xA
9 vex 2554 . . . . . . . . . 10 y V
10 nfcv 2175 . . . . . . . . . 10 xB
11 eqer.1 . . . . . . . . . 10 (x = yA = B)
129, 10, 11csbief 2885 . . . . . . . . 9 y / xA = B
13 csbeq1 2849 . . . . . . . . 9 (y = wy / xA = w / xA)
1412, 13syl5eqr 2083 . . . . . . . 8 (y = wB = w / xA)
1514eqeq2d 2048 . . . . . . 7 (y = w → (A = BA = w / xA))
168, 15sbciegf 2788 . . . . . 6 (w V → ([w / y]A = BA = w / xA))
177, 16ax-mp 7 . . . . 5 ([w / y]A = BA = w / xA)
18 csbeq1a 2854 . . . . . 6 (x = zA = z / xA)
1918eqeq1d 2045 . . . . 5 (x = z → (A = w / xAz / xA = w / xA))
2017, 19syl5bb 181 . . . 4 (x = z → ([w / y]A = Bz / xA = w / xA))
216, 20sbciegf 2788 . . 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 1242   ∈ wcel 1390  Vcvv 2551  [wsbc 2758  ⦋csb 2846   class class class wbr 3755  {copab 3808 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-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 This theorem is referenced by:  eqer  6074
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