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Theorem bj-eleq1w 32040
 Description: Weaker version of eleq1 2676 (but more general than elequ1 1984) not depending on ax-ext 2590 (nor ax-12 2034 nor df-cleq 2603). Remark: this can also be done with eleq1i 2679, eqeltri 2684, eqeltrri 2685, eleq1a 2683, eleq1d 2672, eqeltrd 2688, eqeltrrd 2689, eqneltrd 2707, eqneltrrd 2708, nelneq 2712. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-eleq1w (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))

Proof of Theorem bj-eleq1w
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1940 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21anbi1d 737 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝑥𝑧𝐴) ↔ (𝑧 = 𝑦𝑧𝐴)))
32exbidv 1837 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝑥𝑧𝐴) ↔ ∃𝑧(𝑧 = 𝑦𝑧𝐴)))
4 df-clel 2606 . 2 (𝑥𝐴 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴))
5 df-clel 2606 . 2 (𝑦𝐴 ↔ ∃𝑧(𝑧 = 𝑦𝑧𝐴))
63, 4, 53bitr4g 302 1 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-clel 2606 This theorem is referenced by:  bj-clelsb3  32042  bj-nfcjust  32044
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