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Theorem bj-ax8 32080
 Description: Proof of ax-8 1979 from df-clel 2606 (and FOL). This shows that df-clel 2606 is "too powerful". A possible definition is given by bj-df-clel 32081. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of bj-eleq1w 32040, which has essentially the same proof. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ax8 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Proof of Theorem bj-ax8
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1940 . . . . 5 (𝑥 = 𝑦 → (𝑢 = 𝑥𝑢 = 𝑦))
21anbi1d 737 . . . 4 (𝑥 = 𝑦 → ((𝑢 = 𝑥𝑢𝑧) ↔ (𝑢 = 𝑦𝑢𝑧)))
32exbidv 1837 . . 3 (𝑥 = 𝑦 → (∃𝑢(𝑢 = 𝑥𝑢𝑧) ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧)))
4 df-clel 2606 . . 3 (𝑥𝑧 ↔ ∃𝑢(𝑢 = 𝑥𝑢𝑧))
5 df-clel 2606 . . 3 (𝑦𝑧 ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧))
63, 4, 53bitr4g 302 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
76biimpd 218 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695 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: (None)
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