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Theorem eqerlem 7663
 Description: Lemma for eqer 7664. (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 4911 . 2 (𝑧𝑅𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵)
3 vex 3176 . . 3 𝑧 ∈ V
4 nfcsb1v 3515 . . . . 5 𝑥𝑧 / 𝑥𝐴
5 nfcsb1v 3515 . . . . 5 𝑥𝑤 / 𝑥𝐴
64, 5nfeq 2762 . . . 4 𝑥𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴
7 vex 3176 . . . . . 6 𝑤 ∈ V
8 nfv 1830 . . . . . . 7 𝑦 𝐴 = 𝑤 / 𝑥𝐴
9 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
10 eqer.1 . . . . . . . . . 10 (𝑥 = 𝑦𝐴 = 𝐵)
119, 10csbie 3525 . . . . . . . . 9 𝑦 / 𝑥𝐴 = 𝐵
12 csbeq1 3502 . . . . . . . . 9 (𝑦 = 𝑤𝑦 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1311, 12syl5eqr 2658 . . . . . . . 8 (𝑦 = 𝑤𝐵 = 𝑤 / 𝑥𝐴)
1413eqeq2d 2620 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
158, 14sbciegf 3434 . . . . . 6 (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
167, 15ax-mp 5 . . . . 5 ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴)
17 csbeq1a 3508 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
1817eqeq1d 2612 . . . . 5 (𝑥 = 𝑧 → (𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
1916, 18syl5bb 271 . . . 4 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
206, 19sbciegf 3434 . . 3 (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
213, 20ax-mp 5 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
222, 21bitri 263 1 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499   class class class wbr 4583  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644 This theorem is referenced by:  eqer  7664  eqerOLD  7665
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