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Theorem axrep3 5164
 Description: Axiom of Replacement slightly strengthened from axrep2 5163; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2379. (Revised by BJ, 31-May-2019.)
Assertion
Ref Expression
axrep3 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep3
StepHypRef Expression
1 nfe1 2151 . . . 4 𝑦𝑦𝑧(𝜑𝑧 = 𝑦)
2 nfv 1915 . . . . . 6 𝑦 𝑧𝑥
3 nfv 1915 . . . . . . . 8 𝑦 𝑥𝑤
4 nfa1 2152 . . . . . . . 8 𝑦𝑦𝜑
53, 4nfan 1900 . . . . . . 7 𝑦(𝑥𝑤 ∧ ∀𝑦𝜑)
65nfex 2332 . . . . . 6 𝑦𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)
72, 6nfbi 1904 . . . . 5 𝑦(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
87nfal 2331 . . . 4 𝑦𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
91, 8nfim 1897 . . 3 𝑦(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
109nfex 2332 . 2 𝑦𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
11 axreplem 5162 . 2 (𝑦 = 𝑤 → (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))))
12 axrep2 5163 . 2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
1310, 11, 12chvarfv 2240 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 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 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-rep 5160 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:  axrep4  5165
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