Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfun Structured version   Visualization version   GIF version

Theorem zfun 7096
 Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
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 7095 . 2 𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥)
2 elequ2 2158 . . . . . . 7 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
3 elequ1 2151 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
42, 3anbi12d 608 . . . . . 6 (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧)))
54cbvexv 2434 . . . . 5 (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧))
65imbi1i 338 . . . 4 ((∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
76albii 1894 . . 3 (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87exbii 1923 . 2 (∃𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
91, 8mpbi 220 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∀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-8 2146  ax-9 2153  ax-11 2189  ax-12 2202  ax-13 2407  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857 This theorem is referenced by:  uniex2  7098  axunndlem1  9618
 Copyright terms: Public domain W3C validator