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Theorem zfcndinf 10038
Description: Axiom of Infinity ax-inf 9098, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 5257 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
Assertion
Ref Expression
zfcndinf 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndinf
StepHypRef Expression
1 el 5257 . . 3 𝑤 𝑥𝑤
2 nfv 1916 . . . . . 6 𝑤 𝑥𝑦
3 nfe1 2155 . . . . . . . 8 𝑤𝑤(𝑥𝑤𝑤𝑦)
42, 3nfim 1898 . . . . . . 7 𝑤(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
54nfal 2344 . . . . . 6 𝑤𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
62, 5nfan 1901 . . . . 5 𝑤(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
76nfex 2345 . . . 4 𝑤𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
8 axinfnd 10026 . . . . 5 𝑦(𝑥𝑤 → (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
9819.37iv 1950 . . . 4 (𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
107, 9exlimi 2219 . . 3 (∃𝑤 𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
111, 10ax-mp 5 . 2 𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
12 elequ1 2122 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
13 elequ1 2122 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
1413anbi1d 632 . . . . . . 7 (𝑧 = 𝑥 → ((𝑧𝑤𝑤𝑦) ↔ (𝑥𝑤𝑤𝑦)))
1514exbidv 1923 . . . . . 6 (𝑧 = 𝑥 → (∃𝑤(𝑧𝑤𝑤𝑦) ↔ ∃𝑤(𝑥𝑤𝑤𝑦)))
1612, 15imbi12d 348 . . . . 5 (𝑧 = 𝑥 → ((𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ (𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1716cbvalvw 2044 . . . 4 (∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
1817anbi2i 625 . . 3 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1918exbii 1849 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
2011, 19mpbir 234 1 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-reg 9053  ax-inf 9098
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-nul 4277  df-sn 4551  df-pr 4553
This theorem is referenced by: (None)
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