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Theorem zfcndinf 10656
Description: Axiom of Infinity ax-inf 9676, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5448 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 5448 . . 3 𝑤 𝑥𝑤
2 nfv 1912 . . . . . 6 𝑤 𝑥𝑦
3 nfe1 2148 . . . . . . . 8 𝑤𝑤(𝑥𝑤𝑤𝑦)
42, 3nfim 1894 . . . . . . 7 𝑤(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
54nfal 2322 . . . . . 6 𝑤𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
62, 5nfan 1897 . . . . 5 𝑤(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
76nfex 2323 . . . 4 𝑤𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
8 axinfnd 10644 . . . . 5 𝑦(𝑥𝑤 → (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
9819.37iv 1946 . . . 4 (𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
107, 9exlimi 2215 . . 3 (∃𝑤 𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
111, 10ax-mp 5 . 2 𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
12 elequ1 2113 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
13 elequ1 2113 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
1413anbi1d 631 . . . . . . 7 (𝑧 = 𝑥 → ((𝑧𝑤𝑤𝑦) ↔ (𝑥𝑤𝑤𝑦)))
1514exbidv 1919 . . . . . 6 (𝑧 = 𝑥 → (∃𝑤(𝑧𝑤𝑤𝑦) ↔ ∃𝑤(𝑥𝑤𝑤𝑦)))
1612, 15imbi12d 344 . . . . 5 (𝑧 = 𝑥 → ((𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ (𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1716cbvalvw 2033 . . . 4 (∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
1817anbi2i 623 . . 3 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1918exbii 1845 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
2011, 19mpbir 231 1 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630  ax-inf 9676
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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