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Theorem dfssf 3973
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.) Avoid ax-13 2376. (Revised by GG, 19-May-2023.)
Hypotheses
Ref Expression
dfssf.1 𝑥𝐴
dfssf.2 𝑥𝐵
Assertion
Ref Expression
dfssf (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Proof of Theorem dfssf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ss 3967 . 2 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 dfssf.1 . . . . 5 𝑥𝐴
32nfcri 2896 . . . 4 𝑥 𝑧𝐴
4 dfssf.2 . . . . 5 𝑥𝐵
54nfcri 2896 . . . 4 𝑥 𝑧𝐵
63, 5nfim 1895 . . 3 𝑥(𝑧𝐴𝑧𝐵)
7 nfv 1913 . . 3 𝑧(𝑥𝐴𝑥𝐵)
8 eleq1w 2823 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
9 eleq1w 2823 . . . 4 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
108, 9imbi12d 344 . . 3 (𝑧 = 𝑥 → ((𝑧𝐴𝑧𝐵) ↔ (𝑥𝐴𝑥𝐵)))
116, 7, 10cbvalv1 2342 . 2 (∀𝑧(𝑧𝐴𝑧𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
121, 11bitri 275 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537  wcel 2107  wnfc 2889  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-11 2156  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815  df-nfc 2891  df-ss 3967
This theorem is referenced by:  dfss3f  3974  ssrd  3987  ssrmof  4050  ss2ab  4061  rankval4  9908  rabexgfGS  32519  ballotth  34541  dvcosre  45932  itgsinexplem1  45974
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