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Theorem zfinf 8480
Description: Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfinf 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfinf
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-inf 8479 . 2 𝑥(𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2 elequ1 1994 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
3 elequ1 1994 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
43anbi1d 740 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤𝑧𝑧𝑥) ↔ (𝑦𝑧𝑧𝑥)))
54exbidv 1847 . . . . . 6 (𝑤 = 𝑦 → (∃𝑧(𝑤𝑧𝑧𝑥) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
62, 5imbi12d 334 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
76cbvalv 2272 . . . 4 (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 729 . . 3 ((𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1771 . 2 (∃𝑥(𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
101, 9mpbi 220 1 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
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-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-inf 8479
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  axinf2  8481  axinfndlem1  9371
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