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Theorem ax13dgen2 2170
Description: Degenerate instance of ax-13 2408 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
Assertion
Ref Expression
ax13dgen2 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))

Proof of Theorem ax13dgen2
StepHypRef Expression
1 equcomi 2102 . 2 (𝑦 = 𝑥𝑥 = 𝑦)
2 pm2.21 121 . 2 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑥 𝑦 = 𝑥))
31, 2syl5 34 1 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
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
This theorem is referenced by: (None)
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