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Theorem axrep3 4551
Description: Axiom of Replacement slightly strengthened from axrep2 4550; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.)
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 1968 . . . 4 𝑦𝑦𝑧(𝜑𝑧 = 𝑦)
2 nfv 1792 . . . . . 6 𝑦 𝑧𝑥
3 nfv 1792 . . . . . . . 8 𝑦 𝑥𝑤
4 nfa1 2032 . . . . . . . 8 𝑦𝑦𝜑
53, 4nfan 2064 . . . . . . 7 𝑦(𝑥𝑤 ∧ ∀𝑦𝜑)
65nfex 2083 . . . . . 6 𝑦𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)
72, 6nfbi 2070 . . . . 5 𝑦(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
87nfal 2082 . . . 4 𝑦𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))
91, 8nfim 2056 . . 3 𝑦(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
109nfex 2083 . 2 𝑦𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
11 elequ2 1951 . . . . . . . 8 (𝑦 = 𝑤 → (𝑥𝑦𝑥𝑤))
1211anbi1d 728 . . . . . . 7 (𝑦 = 𝑤 → ((𝑥𝑦 ∧ ∀𝑦𝜑) ↔ (𝑥𝑤 ∧ ∀𝑦𝜑)))
1312exbidv 1799 . . . . . 6 (𝑦 = 𝑤 → (∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑) ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
1413bibi2d 327 . . . . 5 (𝑦 = 𝑤 → ((𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))))
1514albidv 1798 . . . 4 (𝑦 = 𝑤 → (∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)) ↔ ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑))))
1615imbi2d 325 . . 3 (𝑦 = 𝑤 → ((∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ (∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))))
1716exbidv 1799 . 2 (𝑦 = 𝑤 → (∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))))
18 axrep2 4550 . 2 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑)))
1910, 17, 18chvar 2153 1 𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑤 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wa 378  wal 1466  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-rep 4548
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697
This theorem is referenced by:  axrep4  4552
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