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Theorem dfss2f 3559
 Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
dfss2f.1 𝑥𝐴
dfss2f.2 𝑥𝐵
Assertion
Ref Expression
dfss2f (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Proof of Theorem dfss2f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3557 . 2 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 dfss2f.1 . . . . 5 𝑥𝐴
32nfcri 2745 . . . 4 𝑥 𝑧𝐴
4 dfss2f.2 . . . . 5 𝑥𝐵
54nfcri 2745 . . . 4 𝑥 𝑧𝐵
63, 5nfim 1813 . . 3 𝑥(𝑧𝐴𝑧𝐵)
7 nfv 1830 . . 3 𝑧(𝑥𝐴𝑥𝐵)
8 eleq1 2676 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
9 eleq1 2676 . . . 4 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
108, 9imbi12d 333 . . 3 (𝑧 = 𝑥 → ((𝑧𝐴𝑧𝐵) ↔ (𝑥𝐴𝑥𝐵)))
116, 7, 10cbval 2259 . 2 (∀𝑧(𝑧𝐴𝑧𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
121, 11bitri 263 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  Ⅎwnfc 2738   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554 This theorem is referenced by:  dfss3f  3560  ssrd  3573  ss2ab  3633  rankval4  8613  ssrmo  28718  rabexgfGS  28725  ballotth  29926  dvcosre  38799  itgsinexplem1  38845
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