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Theorem bj-ax8 31911
Description: Proof of ax-8 1940 from df-clel 2510 (and FOL). This shows that df-clel 2510 is "too powerful". A possible definition is given by bj-df-clel 31912. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of bj-eleq1w 31871, 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 1903 . . . . 5 (𝑥 = 𝑦 → (𝑢 = 𝑥𝑢 = 𝑦))
21anbi1d 736 . . . 4 (𝑥 = 𝑦 → ((𝑢 = 𝑥𝑢𝑧) ↔ (𝑢 = 𝑦𝑢𝑧)))
32exbidv 1803 . . 3 (𝑥 = 𝑦 → (∃𝑢(𝑢 = 𝑥𝑢𝑧) ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧)))
4 df-clel 2510 . . 3 (𝑥𝑧 ↔ ∃𝑢(𝑢 = 𝑥𝑢𝑧))
5 df-clel 2510 . . 3 (𝑦𝑧 ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧))
63, 4, 53bitr4g 301 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
76biimpd 217 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-clel 2510
This theorem is referenced by: (None)
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