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Theorem axpweq 3915
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3918 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 3364 . . . 4 (𝒫 A V → 𝒫 A 𝒫 𝒫 A)
2 pweq 3354 . . . . . 6 (x = 𝒫 A → 𝒫 x = 𝒫 𝒫 A)
32eleq2d 2104 . . . . 5 (x = 𝒫 A → (𝒫 A 𝒫 x ↔ 𝒫 A 𝒫 𝒫 A))
43spcegv 2635 . . . 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 2560 . . . 4 (𝒫 A 𝒫 x → 𝒫 A V)
76exlimiv 1486 . . 3 (x𝒫 A 𝒫 x → 𝒫 A V)
85, 7impbii 117 . 2 (𝒫 A V ↔ x𝒫 A 𝒫 x)
9 vex 2554 . . . . 5 x V
109elpw2 3902 . . . 4 (𝒫 A 𝒫 x ↔ 𝒫 Ax)
11 pwss 3366 . . . . 5 (𝒫 Axy(yAy x))
12 dfss2 2928 . . . . . . 7 (yAz(z yz A))
1312imbi1i 227 . . . . . 6 ((yAy x) ↔ (z(z yz A) → y x))
1413albii 1356 . . . . 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 1493 . 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 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  wss 2911  𝒫 cpw 3351
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by: (None)
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