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Theorem eleq2w 2202
Description: Weaker version of eleq2 2204 (but more general than elequ2 1692) not depending on ax-ext 2122 nor df-cleq 2133. (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 1692 . . . 4 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21anbi2d 460 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑧𝑥) ↔ (𝑧 = 𝐴𝑧𝑦)))
32exbidv 1798 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴𝑧𝑥) ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦)))
4 df-clel 2136 . 2 (𝐴𝑥 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑥))
5 df-clel 2136 . 2 (𝐴𝑦 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦))
63, 4, 53bitr4g 222 1 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-clel 2136
This theorem is referenced by: (None)
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