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Theorem prtlem11 33598
Description: Lemma for prter2 33613. (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 risset 3060 . . . 4 (𝐶𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐶)
2 r19.41v 3086 . . . . 5 (∃𝑥𝐴 (𝑥 = 𝐶𝐵 = [𝐶] ) ↔ (∃𝑥𝐴 𝑥 = 𝐶𝐵 = [𝐶] ))
3 eceq1 7728 . . . . . . 7 (𝑥 = 𝐶 → [𝑥] = [𝐶] )
4 eqtr3 2647 . . . . . . . 8 (([𝑥] = [𝐶] 𝐵 = [𝐶] ) → [𝑥] = 𝐵)
54eqcomd 2632 . . . . . . 7 (([𝑥] = [𝐶] 𝐵 = [𝐶] ) → 𝐵 = [𝑥] )
63, 5sylan 488 . . . . . 6 ((𝑥 = 𝐶𝐵 = [𝐶] ) → 𝐵 = [𝑥] )
76reximi 3010 . . . . 5 (∃𝑥𝐴 (𝑥 = 𝐶𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
82, 7sylbir 225 . . . 4 ((∃𝑥𝐴 𝑥 = 𝐶𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
91, 8sylanb 489 . . 3 ((𝐶𝐴𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
10 elqsg 7744 . . 3 (𝐵𝐷 → (𝐵 ∈ (𝐴 / ) ↔ ∃𝑥𝐴 𝐵 = [𝑥] ))
119, 10syl5ibr 236 . 2 (𝐵𝐷 → ((𝐶𝐴𝐵 = [𝐶] ) → 𝐵 ∈ (𝐴 / )))
1211expd 452 1 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wrex 2913  [cec 7686   / cqs 7687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ec 7690  df-qs 7694
This theorem is referenced by:  prter2  33613
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