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Theorem prtlem9 33167
 Description: Lemma for prter3 33185. (Contributed by Rodolfo Medina, 25-Sep-2010.)
Assertion
Ref Expression
prtlem9 (𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   (𝑥)

Proof of Theorem prtlem9
StepHypRef Expression
1 risset 3044 . 2 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
2 eceq1 7669 . . 3 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
32reximi 2994 . 2 (∃𝑥𝐵 𝑥 = 𝐴 → ∃𝑥𝐵 [𝑥] = [𝐴] )
41, 3sylbi 206 1 (𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  [cec 7627 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 This theorem is referenced by: (None)
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