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Theorem axpowndlem1 9620
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
Assertion
Ref Expression
axpowndlem1 (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))

Proof of Theorem axpowndlem1
StepHypRef Expression
1 pm2.24 122 . 2 (𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
21sps 2208 1 (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-ex 1852
This theorem is referenced by:  axpownd  9624
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