MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrep1 Structured version   Visualization version   GIF version

Theorem axrep1 4805
Description: The version of the Axiom of Replacement used in the Metamath Solitaire applet http://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 4804 axrep1 4805 axrep2 4806 axrepnd 9454 zfcndrep 9474 = ax-rep 4804. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
axrep1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem axrep1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2044 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑥𝑤𝑥𝑦))
21anbi1d 741 . . . . . . . 8 (𝑤 = 𝑦 → ((𝑥𝑤𝜑) ↔ (𝑥𝑦𝜑)))
32exbidv 1890 . . . . . . 7 (𝑤 = 𝑦 → (∃𝑥(𝑥𝑤𝜑) ↔ ∃𝑥(𝑥𝑦𝜑)))
43bibi2d 331 . . . . . 6 (𝑤 = 𝑦 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑))))
54albidv 1889 . . . . 5 (𝑤 = 𝑦 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑))))
65exbidv 1890 . . . 4 (𝑤 = 𝑦 → (∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)) ↔ ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑))))
76imbi2d 329 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))) ↔ (∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))))
8 ax-rep 4804 . . . 4 (∀𝑥𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
9 19.3v 1954 . . . . . . . 8 (∀𝑦𝜑𝜑)
109imbi1i 338 . . . . . . 7 ((∀𝑦𝜑𝑧 = 𝑦) ↔ (𝜑𝑧 = 𝑦))
1110albii 1787 . . . . . 6 (∀𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑧(𝜑𝑧 = 𝑦))
1211exbii 1814 . . . . 5 (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∃𝑦𝑧(𝜑𝑧 = 𝑦))
1312albii 1787 . . . 4 (∀𝑥𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) ↔ ∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦))
14 nfv 1883 . . . . . . 7 𝑥 𝑧𝑦
15 nfe1 2067 . . . . . . 7 𝑥𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)
1614, 15nfbi 1873 . . . . . 6 𝑥(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
1716nfal 2191 . . . . 5 𝑥𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
18 nfv 1883 . . . . 5 𝑦𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))
19 elequ2 2044 . . . . . . 7 (𝑦 = 𝑥 → (𝑧𝑦𝑧𝑥))
209anbi2i 730 . . . . . . . . 9 ((𝑥𝑤 ∧ ∀𝑦𝜑) ↔ (𝑥𝑤𝜑))
2120exbii 1814 . . . . . . . 8 (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑤𝜑))
2221a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑤𝜑)))
2319, 22bibi12d 334 . . . . . 6 (𝑦 = 𝑥 → ((𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))))
2423albidv 1889 . . . . 5 (𝑦 = 𝑥 → (∀𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑))))
2517, 18, 24cbvex 2308 . . . 4 (∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)) ↔ ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)))
268, 13, 253imtr3i 280 . . 3 (∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤𝜑)))
277, 26chvarv 2299 . 2 (∀𝑥𝑦𝑧(𝜑𝑧 = 𝑦) → ∃𝑥𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
282719.35ri 1847 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-rep 4804
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  axrep2  4806
  Copyright terms: Public domain W3C validator