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Theorem zfun 7442
 Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.) Use ax-un 7441 instead. (New usage is discouraged.)
Assertion
Ref Expression
zfun 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfun
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-un 7441 . 2 𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥)
2 elequ2 2126 . . . . . . 7 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
3 elequ1 2118 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
42, 3anbi12d 633 . . . . . 6 (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧)))
54cbvexvw 2044 . . . . 5 (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧))
65imbi1i 353 . . . 4 ((∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
76albii 1821 . . 3 (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87exbii 1849 . 2 (∃𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
91, 8mpbi 233 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  axunndlem1  10006
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