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Theorem axpweq 4378
 Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4379 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 3813 . . . 4
2 pweq 3804 . . . . . 6
32eleq2d 2505 . . . . 5
43spcegv 3039 . . . 4
51, 4mpd 15 . . 3
6 elex 2966 . . . 4
76exlimiv 1645 . . 3
85, 7impbii 182 . 2
9 vex 2961 . . . . 5
109elpw2 4366 . . . 4
11 pwss 3815 . . . . 5
12 dfss2 3339 . . . . . . 7
1312imbi1i 317 . . . . . 6
1413albii 1576 . . . . 5
1511, 14bitri 242 . . . 4
1610, 15bitri 242 . . 3
1716exbii 1593 . 2
188, 17bitri 242 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178  wal 1550  wex 1551   wceq 1653   wcel 1726  cvv 2958   wss 3322  cpw 3801 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803
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