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Theorem clel5 3654
Description: Alternate definition of class membership: a class 𝑋 is an element of another class 𝐴 iff there is an element of 𝐴 equal to 𝑋. (Contributed by AV, 13-Nov-2020.) Remove use of ax-10 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 19-May-2023.)
Assertion
Ref Expression
clel5 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem clel5
StepHypRef Expression
1 risset 3264 . 2 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑋)
2 eqcom 2825 . . 3 (𝑥 = 𝑋𝑋 = 𝑥)
32rexbii 3244 . 2 (∃𝑥𝐴 𝑥 = 𝑋 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
41, 3bitri 276 1 (𝑋𝐴 ↔ ∃𝑥𝐴 𝑋 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-rex 3141
This theorem is referenced by:  dfss5  4238  4fvwrd4  13015  wrdlen1  13894  phisum  16115  symgmov1  18449  disjunsn  30272
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