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Theorem eleq1w 2713
 Description: Weaker version of eleq1 2718 (but more general than elequ1 2037) not depending on ax-ext 2631 nor df-cleq 2644. (Contributed by BJ, 24-Jun-2019.)
Assertion
Ref Expression
eleq1w (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))

Proof of Theorem eleq1w
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1999 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21anbi1d 741 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝑥𝑧𝐴) ↔ (𝑧 = 𝑦𝑧𝐴)))
32exbidv 1890 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝑥𝑧𝐴) ↔ ∃𝑧(𝑧 = 𝑦𝑧𝐴)))
4 df-clel 2647 . 2 (𝑥𝐴 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴))
5 df-clel 2647 . 2 (𝑦𝐴 ↔ ∃𝑧(𝑧 = 𝑦𝑧𝐴))
63, 4, 53bitr4g 303 1 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∃wex 1744   ∈ wcel 2030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-clel 2647 This theorem is referenced by:  reu8  3435  eqeuel  3974  reuccats1  13526  sumeven  15157  sumodd  15158  numedglnl  26084  uvtxnbgrvtx  26341  wspniunwspnon  26888  fusgr2wsp2nb  27314  numclwlk2lem2f1o  27359  numclwlk2lem2f1oOLD  27366  fsumiunle  29703  frpoinsg  31866  bj-clelsb3  32973  bj-nfcjust  32975  ftc1anclem6  33620  inxprnres  34201  monoordxr  40026  monoord2xr  40028  lmbr3  40297  cnrefiisp  40374  meaiunincf  41018  meaiuninc3v  41019  meaiuninc3  41020  sbgoldbm  41997
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