MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrep4 Structured version   Visualization version   GIF version

Theorem axrep4 5219
Description: A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
axrep4.1 𝑧𝜑
Assertion
Ref Expression
axrep4 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep4
StepHypRef Expression
1 axrep3 5218 . . 3 𝑥(∃𝑧𝑦(𝜑𝑦 = 𝑧) → ∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
2119.35i 1885 . 2 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)))
3 nfv 1921 . . . . 5 𝑧 𝑦𝑥
4 nfv 1921 . . . . . . 7 𝑧 𝑥𝑤
5 nfa1 2152 . . . . . . 7 𝑧𝑧𝜑
64, 5nfan 1906 . . . . . 6 𝑧(𝑥𝑤 ∧ ∀𝑧𝜑)
76nfex 2322 . . . . 5 𝑧𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)
83, 7nfbi 1910 . . . 4 𝑧(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
98nfal 2321 . . 3 𝑧𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑))
10 nfv 1921 . . . . 5 𝑥 𝑦𝑧
11 nfe1 2151 . . . . 5 𝑥𝑥(𝑥𝑤𝜑)
1210, 11nfbi 1910 . . . 4 𝑥(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
1312nfal 2321 . . 3 𝑥𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))
14 elequ2 2125 . . . . 5 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
15 axrep4.1 . . . . . . . . 9 𝑧𝜑
161519.3 2199 . . . . . . . 8 (∀𝑧𝜑𝜑)
1716anbi2i 623 . . . . . . 7 ((𝑥𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥𝑤𝜑))
1817exbii 1854 . . . . . 6 (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑))
1918a1i 11 . . . . 5 (𝑥 = 𝑧 → (∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥𝑤𝜑)))
2014, 19bibi12d 346 . . . 4 (𝑥 = 𝑧 → ((𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
2120albidv 1927 . . 3 (𝑥 = 𝑧 → (∀𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑))))
229, 13, 21cbvexv1 2343 . 2 (∃𝑥𝑦(𝑦𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
232, 22sylib 217 1 (∀𝑥𝑧𝑦(𝜑𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1540  wex 1786  wnf 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-rep 5214
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791
This theorem is referenced by:  axrep5  5220  axprlem3  5352  funimaexg  6518
  Copyright terms: Public domain W3C validator