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Theorem axgroth5 9525
 Description: The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.)
Assertion
Ref Expression
axgroth5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem axgroth5
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ax-groth 9524 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
2 biid 250 . . . 4 (𝑥𝑦𝑥𝑦)
3 pwss 4123 . . . . . 6 (𝒫 𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
4 pwss 4123 . . . . . . 7 (𝒫 𝑧𝑤 ↔ ∀𝑣(𝑣𝑧𝑣𝑤))
54rexbii 3023 . . . . . 6 (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤))
63, 5anbi12i 729 . . . . 5 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
76ralbii 2963 . . . 4 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
8 df-ral 2901 . . . . 5 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)))
9 selpw 4115 . . . . . . 7 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
109imbi1i 338 . . . . . 6 ((𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
1110albii 1737 . . . . 5 (∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
128, 11bitri 263 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
132, 7, 123anbi123i 1244 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))))
1413exbii 1764 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))))
151, 14mpbir 220 1 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-groth 9524 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by:  grothpw  9527  grothpwex  9528  axgroth6  9529  grothtsk  9536
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