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Theorem prtlem11 33169
 Description: Lemma for prter2 33184. (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 3044 . . . 4 (𝐶𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐶)
2 r19.41v 3070 . . . . 5 (∃𝑥𝐴 (𝑥 = 𝐶𝐵 = [𝐶] ) ↔ (∃𝑥𝐴 𝑥 = 𝐶𝐵 = [𝐶] ))
3 eceq1 7669 . . . . . . 7 (𝑥 = 𝐶 → [𝑥] = [𝐶] )
4 eqtr3 2631 . . . . . . . 8 (([𝑥] = [𝐶] 𝐵 = [𝐶] ) → [𝑥] = 𝐵)
54eqcomd 2616 . . . . . . 7 (([𝑥] = [𝐶] 𝐵 = [𝐶] ) → 𝐵 = [𝑥] )
63, 5sylan 487 . . . . . 6 ((𝑥 = 𝐶𝐵 = [𝐶] ) → 𝐵 = [𝑥] )
76reximi 2994 . . . . 5 (∃𝑥𝐴 (𝑥 = 𝐶𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
82, 7sylbir 224 . . . 4 ((∃𝑥𝐴 𝑥 = 𝐶𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
91, 8sylanb 488 . . 3 ((𝐶𝐴𝐵 = [𝐶] ) → ∃𝑥𝐴 𝐵 = [𝑥] )
10 elqsg 7685 . . 3 (𝐵𝐷 → (𝐵 ∈ (𝐴 / ) ↔ ∃𝑥𝐴 𝐵 = [𝑥] ))
119, 10syl5ibr 235 . 2 (𝐵𝐷 → ((𝐶𝐴𝐵 = [𝐶] ) → 𝐵 ∈ (𝐴 / )))
1211expd 451 1 (𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  [cec 7627   / cqs 7628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631  df-qs 7635 This theorem is referenced by:  prter2  33184
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