MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrep2 Structured version   Visualization version   GIF version

Theorem axrep2 4550
Description: Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.)
Assertion
Ref Expression
axrep2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axrep2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfe1 1968 . . . . 5 𝑤𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤)
2 nfv 1792 . . . . 5 𝑤𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))
31, 2nfim 2056 . . . 4 𝑤(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
43nfex 2083 . . 3 𝑤𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
5 elequ2 1951 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
65anbi1d 728 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑥𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥𝑦 ∧ ∀𝑦𝜑)))
76exbidv 1799 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
87bibi2d 327 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
98albidv 1798 . . . . 5 (𝑤 = 𝑦 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
109imbi2d 325 . . . 4 (𝑤 = 𝑦 → ((∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))) ↔ (∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
1110exbidv 1799 . . 3 (𝑤 = 𝑦 → (∃𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))))
12 axrep1 4549 . . 3 𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
134, 11, 12chvar 2153 . 2 𝑥(∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
14 sp 1990 . . . . . . 7 (∀𝑦𝜑𝜑)
1514imim1i 60 . . . . . 6 ((𝜑𝑧 = 𝑦) → (∀𝑦𝜑𝑧 = 𝑦))
1615alimi 1713 . . . . 5 (∀𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦𝜑𝑧 = 𝑦))
1716eximi 1738 . . . 4 (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦))
18 nfv 1792 . . . . 5 𝑤𝑧(∀𝑦𝜑𝑧 = 𝑦)
19 nfa1 2032 . . . . . . 7 𝑦𝑦𝜑
20 nfv 1792 . . . . . . 7 𝑦 𝑧 = 𝑤
2119, 20nfim 2056 . . . . . 6 𝑦(∀𝑦𝜑𝑧 = 𝑤)
2221nfal 2082 . . . . 5 𝑦𝑧(∀𝑦𝜑𝑧 = 𝑤)
23 equequ2 1902 . . . . . . 7 (𝑦 = 𝑤 → (𝑧 = 𝑦𝑧 = 𝑤))
2423imbi2d 325 . . . . . 6 (𝑦 = 𝑤 → ((∀𝑦𝜑𝑧 = 𝑦) ↔ (∀𝑦𝜑𝑧 = 𝑤)))
2524albidv 1798 . . . . 5 (𝑦 = 𝑤 → (∀𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦𝜑𝑧 = 𝑤)))
2618, 22, 25cbvex 2162 . . . 4 (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤))
2717, 26sylib 203 . . 3 (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤))
2827imim1i 60 . 2 ((∃𝑤𝑧(∀𝑦𝜑𝑧 = 𝑤) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) → (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))
2913, 28eximii 1740 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wa 378  wal 1466  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-rep 4548
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697
This theorem is referenced by:  axrep3  4551  axrepndlem1  9102
  Copyright terms: Public domain W3C validator