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Theorem zfinf 8574
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 8573 . 2 𝑥(𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)))
2 elequ1 2037 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝑥𝑦𝑥))
3 elequ1 2037 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
43anbi1d 741 . . . . . . 7 (𝑤 = 𝑦 → ((𝑤𝑧𝑧𝑥) ↔ (𝑦𝑧𝑧𝑥)))
54exbidv 1890 . . . . . 6 (𝑤 = 𝑦 → (∃𝑧(𝑤𝑧𝑧𝑥) ↔ ∃𝑧(𝑦𝑧𝑧𝑥)))
62, 5imbi12d 333 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ (𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
76cbvalv 2309 . . . 4 (∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥)) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
87anbi2i 730 . . 3 ((𝑦𝑥 ∧ ∀𝑤(𝑤𝑥 → ∃𝑧(𝑤𝑧𝑧𝑥))) ↔ (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))
98exbii 1814 . 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 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-11 2074  ax-12 2087  ax-13 2282  ax-inf 8573
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  axinf2  8575  axinfndlem1  9465
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