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Theorem prtlem9 35531
Description: Lemma for prter3 35549. (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 3231 . 2 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
2 eceq1 8177 . . 3 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
32reximi 3207 . 2 (∃𝑥𝐵 𝑥 = 𝐴 → ∃𝑥𝐵 [𝑥] = [𝐴] )
41, 3sylbi 218 1 (𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  wcel 2081  wrex 3106  [cec 8137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ec 8141
This theorem is referenced by: (None)
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