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Theorem zfinf 9086
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 9085 . 2 𝑥(𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2 elequ1 2122 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
3 elequ1 2122 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
43anbi1d 632 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤𝑧𝑧𝑥) ↔ (𝑦𝑧𝑧𝑥)))
54exbidv 1923 . . . . . 6 (𝑤 = 𝑦 → (∃𝑧(𝑤𝑧𝑧𝑥) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
62, 5imbi12d 348 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
76cbvalvw 2044 . . . 4 (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 625 . . 3 ((𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1849 . 2 (∃𝑥(𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
101, 9mpbi 233 1 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wex 1781
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-inf 9085
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  axinf2  9087  axinfndlem1  10012
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