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Theorem eqerlem 6165
Description: Lemma for eqer 6166. (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 4023 . 2 (𝑧𝑅𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵)
3 vex 2575 . . 3 𝑧 ∈ V
4 nfcsb1v 2907 . . . . 5 𝑥𝑧 / 𝑥𝐴
5 nfcsb1v 2907 . . . . 5 𝑥𝑤 / 𝑥𝐴
64, 5nfeq 2199 . . . 4 𝑥𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴
7 vex 2575 . . . . . 6 𝑤 ∈ V
8 nfv 1435 . . . . . . 7 𝑦 𝐴 = 𝑤 / 𝑥𝐴
9 vex 2575 . . . . . . . . . 10 𝑦 ∈ V
10 nfcv 2192 . . . . . . . . . 10 𝑥𝐵
11 eqer.1 . . . . . . . . . 10 (𝑥 = 𝑦𝐴 = 𝐵)
129, 10, 11csbief 2916 . . . . . . . . 9 𝑦 / 𝑥𝐴 = 𝐵
13 csbeq1 2880 . . . . . . . . 9 (𝑦 = 𝑤𝑦 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1412, 13syl5eqr 2100 . . . . . . . 8 (𝑦 = 𝑤𝐵 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2065 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
168, 15sbciegf 2814 . . . . . 6 (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
177, 16ax-mp 7 . . . . 5 ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴)
18 csbeq1a 2885 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
1918eqeq1d 2062 . . . . 5 (𝑥 = 𝑧 → (𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
2017, 19syl5bb 185 . . . 4 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
216, 20sbciegf 2814 . . 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 177 1 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1257  wcel 1407  Vcvv 2572  [wsbc 2784  csb 2877   class class class wbr 3789  {copab 3842
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 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574  df-sbc 2785  df-csb 2878  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844
This theorem is referenced by:  eqer  6166
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