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Theorem axpweq 4629
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4630 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
Hypothesis
Ref Expression
axpweq.1
Assertion
Ref Expression
axpweq
Distinct variable group:   , , ,

Proof of Theorem axpweq
StepHypRef Expression
1 pwidg 4025 . . . 4
2 pweq 4015 . . . . . 6
32eleq2d 2527 . . . . 5
43spcegv 3195 . . . 4
51, 4mpd 15 . . 3
6 elex 3118 . . . 4
76exlimiv 1722 . . 3
85, 7impbii 188 . 2
9 vex 3112 . . . . 5
109elpw2 4616 . . . 4
11 pwss 4027 . . . . 5
12 dfss2 3492 . . . . . . 7
1312imbi1i 325 . . . . . 6
1413albii 1640 . . . . 5
1511, 14bitri 249 . . . 4
1610, 15bitri 249 . . 3
1716exbii 1667 . 2
188, 17bitri 249 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  C_wss 3475  ~Pcpw 4012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014
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