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Theorem prtlem9 36615
Description: Lemma for prter3 36633. (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 3186 . 2 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
2 eceq1 8429 . . 3 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
32reximi 3166 . 2 (∃𝑥𝐵 𝑥 = 𝐴 → ∃𝑥𝐵 [𝑥] = [𝐴] )
41, 3sylbi 220 1 (𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wrex 3062  [cec 8389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ec 8393
This theorem is referenced by: (None)
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