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Theorem bj-ax8 33216
Description: Proof of ax-8 2147 from df-clel 2767 (and FOL). This shows that df-clel 2767 is "too powerful". A possible definition is given by bj-df-clel 33217. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of eleq1w 2833, 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 2111 . . . . 5 (𝑥 = 𝑦 → (𝑢 = 𝑥𝑢 = 𝑦))
21anbi1d 615 . . . 4 (𝑥 = 𝑦 → ((𝑢 = 𝑥𝑢𝑧) ↔ (𝑢 = 𝑦𝑢𝑧)))
32exbidv 2002 . . 3 (𝑥 = 𝑦 → (∃𝑢(𝑢 = 𝑥𝑢𝑧) ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧)))
4 df-clel 2767 . . 3 (𝑥𝑧 ↔ ∃𝑢(𝑢 = 𝑥𝑢𝑧))
5 df-clel 2767 . . 3 (𝑦𝑧 ↔ ∃𝑢(𝑢 = 𝑦𝑢𝑧))
63, 4, 53bitr4g 303 . 2 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
76biimpd 219 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-clel 2767
This theorem is referenced by: (None)
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