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Theorem eleq2w 2714
Description: Weaker version of eleq2 2719 (but more general than elequ2 2044) not depending on ax-ext 2631 nor df-cleq 2644. (Contributed by BJ, 29-Sep-2019.)
Assertion
Ref Expression
eleq2w (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))

Proof of Theorem eleq2w
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2044 . . . 4 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21anbi2d 740 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑧𝑥) ↔ (𝑧 = 𝐴𝑧𝑦)))
32exbidv 1890 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴𝑧𝑥) ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦)))
4 df-clel 2647 . 2 (𝐴𝑥 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑥))
5 df-clel 2647 . 2 (𝐴𝑦 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦))
63, 4, 53bitr4g 303 1 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  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  ax-9 2039
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-clel 2647
This theorem is referenced by:  usgredgleordALT  26171  vtxdushgrfvedglem  26441  lmbr3  40297
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