Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrep5 Structured version   Visualization version   GIF version

Theorem axrep5 5182
 Description: Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 𝜑 is analogous to a "function" from 𝑥 to 𝑦 (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set 𝑧 that corresponds to the "image" of 𝜑 restricted to some other set 𝑤. The hypothesis says 𝑧 must not be free in 𝜑. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
axrep5.1 𝑧𝜑
Assertion
Ref Expression
axrep5 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep5
StepHypRef Expression
1 19.37v 1999 . . . . 5 (∃𝑧(𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ (𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)))
2 impexp 454 . . . . . . . 8 (((𝑥𝑤𝜑) → 𝑦 = 𝑧) ↔ (𝑥𝑤 → (𝜑𝑦 = 𝑧)))
32albii 1821 . . . . . . 7 (∀𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧) ↔ ∀𝑦(𝑥𝑤 → (𝜑𝑦 = 𝑧)))
4 19.21v 1941 . . . . . . 7 (∀𝑦(𝑥𝑤 → (𝜑𝑦 = 𝑧)) ↔ (𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)))
53, 4bitr2i 279 . . . . . 6 ((𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ ∀𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
65exbii 1849 . . . . 5 (∃𝑧(𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ ∃𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
71, 6bitr3i 280 . . . 4 ((𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) ↔ ∃𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
87albii 1821 . . 3 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) ↔ ∀𝑥𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
9 nfv 1916 . . . . 5 𝑧 𝑥𝑤
10 axrep5.1 . . . . 5 𝑧𝜑
119, 10nfan 1901 . . . 4 𝑧(𝑥𝑤𝜑)
1211axrep4 5181 . . 3 (∀𝑥𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))))
138, 12sylbi 220 . 2 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))))
14 anabs5 662 . . . . . 6 ((𝑥𝑤 ∧ (𝑥𝑤𝜑)) ↔ (𝑥𝑤𝜑))
1514exbii 1849 . . . . 5 (∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑)) ↔ ∃𝑥(𝑥𝑤𝜑))
1615bibi2i 341 . . . 4 ((𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1716albii 1821 . . 3 (∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1817exbii 1849 . 2 (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1913, 18sylib 221 1 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 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 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-rep 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  zfrepclf  5184
 Copyright terms: Public domain W3C validator