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Theorem bj-eleq2w 32041
 Description: Weaker version of eleq2 2677 (but more general than elequ2 1991) not depending on ax-ext 2590 (nor ax-12 2034 nor df-cleq 2603). (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-eleq2w (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))

Proof of Theorem bj-eleq2w
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elequ2 1991 . . . 4 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21anbi2d 736 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝐴𝑧𝑥) ↔ (𝑧 = 𝐴𝑧𝑦)))
32exbidv 1837 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝐴𝑧𝑥) ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦)))
4 df-clel 2606 . 2 (𝐴𝑥 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑥))
5 df-clel 2606 . 2 (𝐴𝑦 ↔ ∃𝑧(𝑧 = 𝐴𝑧𝑦))
63, 4, 53bitr4g 302 1 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃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  ax-9 1986 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-clel 2606 This theorem is referenced by: (None)
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