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Theorem eqerlem 8309
 Description: Lemma for eqer 8310. (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 5384 . 2 (𝑧𝑅𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵)
3 nfcsb1v 3852 . . . . 5 𝑥𝑧 / 𝑥𝐴
4 nfcsb1v 3852 . . . . 5 𝑥𝑤 / 𝑥𝐴
53, 4nfeq 2968 . . . 4 𝑥𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴
6 nfv 1915 . . . . . . 7 𝑦 𝐴 = 𝑤 / 𝑥𝐴
7 vex 3444 . . . . . . . . . 10 𝑦 ∈ V
8 eqer.1 . . . . . . . . . 10 (𝑥 = 𝑦𝐴 = 𝐵)
97, 8csbie 3863 . . . . . . . . 9 𝑦 / 𝑥𝐴 = 𝐵
10 csbeq1 3831 . . . . . . . . 9 (𝑦 = 𝑤𝑦 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
119, 10syl5eqr 2847 . . . . . . . 8 (𝑦 = 𝑤𝐵 = 𝑤 / 𝑥𝐴)
1211eqeq2d 2809 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
136, 12sbciegf 3757 . . . . . 6 (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
1413elv 3446 . . . . 5 ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴)
15 csbeq1a 3842 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
1615eqeq1d 2800 . . . . 5 (𝑥 = 𝑧 → (𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
1714, 16syl5bb 286 . . . 4 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
185, 17sbciegf 3757 . . 3 (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
1918elv 3446 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
202, 19bitri 278 1 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  Vcvv 3441  [wsbc 3720  ⦋csb 3828   class class class wbr 5031  {copab 5093 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094 This theorem is referenced by:  eqer  8310
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