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Theorem zfcndinf 9553
Description: Axiom of Infinity ax-inf 8648, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4952 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 4952 . . 3 𝑤 𝑥𝑤
2 nfv 1956 . . . . . 6 𝑤 𝑥𝑦
3 nfe1 2140 . . . . . . . 8 𝑤𝑤(𝑥𝑤𝑤𝑦)
42, 3nfim 1938 . . . . . . 7 𝑤(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
54nfal 2264 . . . . . 6 𝑤𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
62, 5nfan 1941 . . . . 5 𝑤(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
76nfex 2265 . . . 4 𝑤𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
8 axinfnd 9541 . . . . 5 𝑦(𝑥𝑤 → (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
9819.37iv 1989 . . . 4 (𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
107, 9exlimi 2197 . . 3 (∃𝑤 𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
111, 10ax-mp 5 . 2 𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
12 elequ1 2110 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
13 elequ1 2110 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
1413anbi1d 743 . . . . . . 7 (𝑧 = 𝑥 → ((𝑧𝑤𝑤𝑦) ↔ (𝑥𝑤𝑤𝑦)))
1514exbidv 1963 . . . . . 6 (𝑧 = 𝑥 → (∃𝑤(𝑧𝑤𝑤𝑦) ↔ ∃𝑤(𝑥𝑤𝑤𝑦)))
1612, 15imbi12d 333 . . . . 5 (𝑧 = 𝑥 → ((𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ (𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1716cbvalv 2382 . . . 4 (∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
1817anbi2i 732 . . 3 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1918exbii 1887 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
2011, 19mpbir 221 1 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1594   = wceq 1596  wex 1817  wcel 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-reg 8613  ax-inf 8648
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-v 3306  df-dif 3683  df-un 3685  df-nul 4024  df-sn 4286  df-pr 4288
This theorem is referenced by: (None)
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