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Theorem axrep5 4698
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 1896 . . . . 5 (∃𝑧(𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ (𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)))
2 impexp 460 . . . . . . . 8 (((𝑥𝑤𝜑) → 𝑦 = 𝑧) ↔ (𝑥𝑤 → (𝜑𝑦 = 𝑧)))
32albii 1736 . . . . . . 7 (∀𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧) ↔ ∀𝑦(𝑥𝑤 → (𝜑𝑦 = 𝑧)))
4 19.21v 1854 . . . . . . 7 (∀𝑦(𝑥𝑤 → (𝜑𝑦 = 𝑧)) ↔ (𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)))
53, 4bitr2i 263 . . . . . 6 ((𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ ∀𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
65exbii 1763 . . . . 5 (∃𝑧(𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ ∃𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
71, 6bitr3i 264 . . . 4 ((𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) ↔ ∃𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
87albii 1736 . . 3 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) ↔ ∀𝑥𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
9 nfv 1829 . . . . 5 𝑧 𝑥𝑤
10 axrep5.1 . . . . 5 𝑧𝜑
119, 10nfan 1815 . . . 4 𝑧(𝑥𝑤𝜑)
1211axrep4 4697 . . 3 (∀𝑥𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))))
138, 12sylbi 205 . 2 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))))
14 anabs5 846 . . . . . 6 ((𝑥𝑤 ∧ (𝑥𝑤𝜑)) ↔ (𝑥𝑤𝜑))
1514exbii 1763 . . . . 5 (∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑)) ↔ ∃𝑥(𝑥𝑤𝜑))
1615bibi2i 325 . . . 4 ((𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1716albii 1736 . . 3 (∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1817exbii 1763 . 2 (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1913, 18sylib 206 1 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-rep 4693
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by:  zfrepclf  4699  axsep  4702
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