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Theorem axgroth5 9598
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 9597 . 2 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
2 biid 251 . . . 4 (𝑥𝑦𝑥𝑦)
3 pwss 4151 . . . . . 6 (𝒫 𝑧𝑦 ↔ ∀𝑤(𝑤𝑧𝑤𝑦))
4 pwss 4151 . . . . . . 7 (𝒫 𝑧𝑤 ↔ ∀𝑣(𝑣𝑧𝑣𝑤))
54rexbii 3035 . . . . . 6 (∃𝑤𝑦 𝒫 𝑧𝑤 ↔ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤))
63, 5anbi12i 732 . . . . 5 ((𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
76ralbii 2975 . . . 4 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ↔ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)))
8 df-ral 2912 . . . . 5 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)))
9 selpw 4142 . . . . . . 7 (𝑧 ∈ 𝒫 𝑦𝑧𝑦)
109imbi1i 339 . . . . . 6 ((𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ (𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
1110albii 1744 . . . . 5 (∀𝑧(𝑧 ∈ 𝒫 𝑦 → (𝑧𝑦𝑧𝑦)) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
128, 11bitri 264 . . . 4 (∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦) ↔ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦)))
132, 7, 123anbi123i 1249 . . 3 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) ↔ (𝑥𝑦 ∧ ∀𝑧𝑦 (∀𝑤(𝑤𝑧𝑤𝑦) ∧ ∃𝑤𝑦𝑣(𝑣𝑧𝑣𝑤)) ∧ ∀𝑧(𝑧𝑦 → (𝑧𝑦𝑧𝑦))))
1413exbii 1771 . 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 221 1 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1036  wal 1478  wex 1701  wcel 1987  wral 2907  wrex 2908  wss 3559  𝒫 cpw 4135   class class class wbr 4618  cen 7904
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-groth 9597
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-in 3566  df-ss 3573  df-pw 4137
This theorem is referenced by:  grothpw  9600  grothpwex  9601  axgroth6  9602  grothtsk  9609
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