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Theorem prtlem9 37376
Description: Lemma for prter3 37394. (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 3220 . 2 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
2 eceq1 8692 . . 3 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
32reximi 3084 . 2 (∃𝑥𝐵 𝑥 = 𝐴 → ∃𝑥𝐵 [𝑥] = [𝐴] )
41, 3sylbi 216 1 (𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wrex 3070  [cec 8652
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ec 8656
This theorem is referenced by: (None)
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