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Theorem eleq2w 2239
Description: Weaker version of eleq2 2241 (but more general than elequ2 2153) not depending on ax-ext 2159 nor df-cleq 2170. (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 2153 . . . 4 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21anbi2d 464 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑧𝑥) ↔ (𝑧 = 𝐴𝑧𝑦)))
32exbidv 1825 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴𝑧𝑥) ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦)))
4 df-clel 2173 . 2 (𝐴𝑥 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑥))
5 df-clel 2173 . 2 (𝐴𝑦 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦))
63, 4, 53bitr4g 223 1 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-14 2151
This theorem depends on definitions:  df-bi 117  df-clel 2173
This theorem is referenced by:  exmidontriimlem4  7217
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