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Theorem prtlem9 34820
Description: Lemma for prter3 34838. (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 3209 . 2 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
2 eceq1 7985 . . 3 (𝑥 = 𝐴 → [𝑥] = [𝐴] )
32reximi 3157 . 2 (∃𝑥𝐵 𝑥 = 𝐴 → ∃𝑥𝐵 [𝑥] = [𝐴] )
41, 3sylbi 208 1 (𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  wrex 3056  [cec 7945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ec 7949
This theorem is referenced by: (None)
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