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Theorem zfcndrep 9396
Description: Axiom of Replacement ax-rep 4741, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
Assertion
Ref Expression
zfcndrep (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem zfcndrep
StepHypRef Expression
1 nfe1 2024 . . . . . 6 𝑦𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦)
2 nfv 1840 . . . . . . . 8 𝑦 𝑧𝑤
3 nfv 1840 . . . . . . . . . 10 𝑦 𝑤𝑥
4 nfa1 2025 . . . . . . . . . 10 𝑦𝑦𝑦𝜑
53, 4nfan 1825 . . . . . . . . 9 𝑦(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)
65nfex 2151 . . . . . . . 8 𝑦𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)
72, 6nfbi 1830 . . . . . . 7 𝑦(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑))
87nfal 2150 . . . . . 6 𝑦𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑))
91, 8nfim 1822 . . . . 5 𝑦(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))
109nfex 2151 . . . 4 𝑦𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))
11 elequ2 2001 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑤𝑦𝑤𝑥))
1211anbi1d 740 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑤𝑦 ∧ ∀𝑦𝑦𝜑) ↔ (𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))
1312exbidv 1847 . . . . . . . 8 (𝑦 = 𝑥 → (∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑) ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))
1413bibi2d 332 . . . . . . 7 (𝑦 = 𝑥 → ((𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑)) ↔ (𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑))))
1514albidv 1846 . . . . . 6 (𝑦 = 𝑥 → (∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑)) ↔ ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑))))
1615imbi2d 330 . . . . 5 (𝑦 = 𝑥 → ((∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑))) ↔ (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))))
1716exbidv 1847 . . . 4 (𝑦 = 𝑥 → (∃𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑))) ↔ ∃𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))))
18 axrepnd 9376 . . . . 5 𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑤 ↔ ∃𝑤(∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑)))
19 19.3v 1894 . . . . . . . . 9 (∀𝑦 𝑧𝑤𝑧𝑤)
20 19.3v 1894 . . . . . . . . . . 11 (∀𝑧 𝑤𝑦𝑤𝑦)
2120anbi1i 730 . . . . . . . . . 10 ((∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑) ↔ (𝑤𝑦 ∧ ∀𝑦𝑦𝜑))
2221exbii 1771 . . . . . . . . 9 (∃𝑤(∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑) ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑))
2319, 22bibi12i 329 . . . . . . . 8 ((∀𝑦 𝑧𝑤 ↔ ∃𝑤(∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑)) ↔ (𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑)))
2423albii 1744 . . . . . . 7 (∀𝑧(∀𝑦 𝑧𝑤 ↔ ∃𝑤(∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑)) ↔ ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑)))
2524imbi2i 326 . . . . . 6 ((∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑤 ↔ ∃𝑤(∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑))) ↔ (∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑))))
2625exbii 1771 . . . . 5 (∃𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑤 ↔ ∃𝑤(∀𝑧 𝑤𝑦 ∧ ∀𝑦𝑦𝜑))) ↔ ∃𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑))))
2718, 26mpbi 220 . . . 4 𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑦 ∧ ∀𝑦𝑦𝜑)))
2810, 17, 27chvar 2261 . . 3 𝑤(∃𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))
292819.35i 1803 . 2 (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑤𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)))
30 nfv 1840 . . . . 5 𝑤 𝑧𝑦
31 nfe1 2024 . . . . 5 𝑤𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)
3230, 31nfbi 1830 . . . 4 𝑤(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
3332nfal 2150 . . 3 𝑤𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
34 elequ2 2001 . . . . 5 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
35 nfa1 2025 . . . . . . . . 9 𝑦𝑦𝜑
363519.3 2067 . . . . . . . 8 (∀𝑦𝑦𝜑 ↔ ∀𝑦𝜑)
3736anbi2i 729 . . . . . . 7 ((𝑤𝑥 ∧ ∀𝑦𝑦𝜑) ↔ (𝑤𝑥 ∧ ∀𝑦𝜑))
3837exbii 1771 . . . . . 6 (∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑) ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))
3938a1i 11 . . . . 5 (𝑤 = 𝑦 → (∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑) ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
4034, 39bibi12d 335 . . . 4 (𝑤 = 𝑦 → ((𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)) ↔ (𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))))
4140albidv 1846 . . 3 (𝑤 = 𝑦 → (∀𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)) ↔ ∀𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑))))
428, 33, 41cbvex 2271 . 2 (∃𝑤𝑧(𝑧𝑤 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝑦𝜑)) ↔ ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
4329, 42sylib 208 1 (∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-reg 8457
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-un 3565  df-nul 3898  df-sn 4156  df-pr 4158
This theorem is referenced by: (None)
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