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Theorem prtlem11 37259
Description: Lemma for prter2 37274. (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 8620 . . . 4 (𝑥 = 𝐶 → [𝑥] = [𝐶] )
21rspceeqv 3594 . . 3 ((𝐶𝐴𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
3 elqsg 8641 . . 3 (𝐵𝐷 → (𝐵 ∈ (𝐴 / ) ↔ ∃𝑥𝐴 𝐵 = [𝑥] ))
42, 3syl5ibr 246 . 2 (𝐵𝐷 → ((𝐶𝐴𝐵 = [𝐶] ) → 𝐵 ∈ (𝐴 / )))
54expd 417 1 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3072  [cec 8580   / cqs 8581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8584  df-qs 8588
This theorem is referenced by:  prter2  37274
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