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Theorem eqerlem 7930
Description: Lemma for eqer 7931. (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 5119 . 2 (𝑧𝑅𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵)
3 vex 3354 . . 3 𝑧 ∈ V
4 nfcsb1v 3698 . . . . 5 𝑥𝑧 / 𝑥𝐴
5 nfcsb1v 3698 . . . . 5 𝑥𝑤 / 𝑥𝐴
64, 5nfeq 2925 . . . 4 𝑥𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴
7 vex 3354 . . . . . 6 𝑤 ∈ V
8 nfv 1995 . . . . . . 7 𝑦 𝐴 = 𝑤 / 𝑥𝐴
9 vex 3354 . . . . . . . . . 10 𝑦 ∈ V
10 eqer.1 . . . . . . . . . 10 (𝑥 = 𝑦𝐴 = 𝐵)
119, 10csbie 3708 . . . . . . . . 9 𝑦 / 𝑥𝐴 = 𝐵
12 csbeq1 3685 . . . . . . . . 9 (𝑦 = 𝑤𝑦 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1311, 12syl5eqr 2819 . . . . . . . 8 (𝑦 = 𝑤𝐵 = 𝑤 / 𝑥𝐴)
1413eqeq2d 2781 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
158, 14sbciegf 3619 . . . . . 6 (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
167, 15ax-mp 5 . . . . 5 ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴)
17 csbeq1a 3691 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
1817eqeq1d 2773 . . . . 5 (𝑥 = 𝑧 → (𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
1916, 18syl5bb 272 . . . 4 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
206, 19sbciegf 3619 . . 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 264 1 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  [wsbc 3587  csb 3682   class class class wbr 4786  {copab 4846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847
This theorem is referenced by:  eqer  7931
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