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Theorem axrep3 5191
Description: Axiom of Replacement slightly strengthened from axrep2 5190; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2385. (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 2147 . . . 4 𝑦𝑦𝑧(𝜑𝑧 = 𝑦)
2 nfv 1908 . . . . . 6 𝑦 𝑧𝑥
3 nfv 1908 . . . . . . . 8 𝑦 𝑥𝑤
4 nfa1 2148 . . . . . . . 8 𝑦𝑦𝜑
53, 4nfan 1893 . . . . . . 7 𝑦(𝑥𝑤 ∧ ∀𝑦𝜑)
65nfex 2337 . . . . . 6 𝑦𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)
72, 6nfbi 1897 . . . . 5 𝑦(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
87nfal 2336 . . . 4 𝑦𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
91, 8nfim 1890 . . 3 𝑦(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
109nfex 2337 . 2 𝑦𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
11 axreplem 5189 . 2 (𝑦 = 𝑤 → (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))))
12 axrep2 5190 . 2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
1310, 11, 12chvarfv 2235 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-rep 5187
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778
This theorem is referenced by:  axrep4  5192
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