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Theorem axpweq 3898
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3901 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
Hypothesis
Ref Expression
axpweq.1 A V
Assertion
Ref Expression
axpweq (𝒫 A V ↔ xy(z(z yz A) → y x))
Distinct variable group:   x,y,z,A

Proof of Theorem axpweq
StepHypRef Expression
1 pwidg 3347 . . . 4 (𝒫 A V → 𝒫 A 𝒫 𝒫 A)
2 pweq 3337 . . . . . 6 (x = 𝒫 A → 𝒫 x = 𝒫 𝒫 A)
32eleq2d 2089 . . . . 5 (x = 𝒫 A → (𝒫 A 𝒫 x ↔ 𝒫 A 𝒫 𝒫 A))
43spcegv 2618 . . . 4 (𝒫 A V → (𝒫 A 𝒫 𝒫 Ax𝒫 A 𝒫 x))
51, 4mpd 13 . . 3 (𝒫 A V → x𝒫 A 𝒫 x)
6 elex 2543 . . . 4 (𝒫 A 𝒫 x → 𝒫 A V)
76exlimiv 1471 . . 3 (x𝒫 A 𝒫 x → 𝒫 A V)
85, 7impbii 117 . 2 (𝒫 A V ↔ x𝒫 A 𝒫 x)
9 vex 2538 . . . . 5 x V
109elpw2 3885 . . . 4 (𝒫 A 𝒫 x ↔ 𝒫 Ax)
11 pwss 3349 . . . . 5 (𝒫 Axy(yAy x))
12 dfss2 2911 . . . . . . 7 (yAz(z yz A))
1312imbi1i 227 . . . . . 6 ((yAy x) ↔ (z(z yz A) → y x))
1413albii 1339 . . . . 5 (y(yAy x) ↔ y(z(z yz A) → y x))
1511, 14bitri 173 . . . 4 (𝒫 Axy(z(z yz A) → y x))
1610, 15bitri 173 . . 3 (𝒫 A 𝒫 xy(z(z yz A) → y x))
1716exbii 1478 . 2 (x𝒫 A 𝒫 xxy(z(z yz A) → y x))
188, 17bitri 173 1 (𝒫 A V ↔ xy(z(z yz A) → y x))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228  wex 1362   wcel 1374  Vcvv 2535  wss 2894  𝒫 cpw 3334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336
This theorem is referenced by: (None)
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