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Theorem prtlem11 35882
Description: Lemma for prter2 35897. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
prtlem11 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))

Proof of Theorem prtlem11
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eceq1 8316 . . . 4 (𝑥 = 𝐶 → [𝑥] = [𝐶] )
21rspceeqv 3635 . . 3 ((𝐶𝐴𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
3 elqsg 8337 . . 3 (𝐵𝐷 → (𝐵 ∈ (𝐴 / ) ↔ ∃𝑥𝐴 𝐵 = [𝑥] ))
42, 3syl5ibr 247 . 2 (𝐵𝐷 → ((𝐶𝐴𝐵 = [𝐶] ) → 𝐵 ∈ (𝐴 / )))
54expd 416 1 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  [cec 8276   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-qs 8284
This theorem is referenced by:  prter2  35897
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