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Theorem dfssf 3986
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 2375. (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 3980 . 2 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 dfssf.1 . . . . 5 𝑥𝐴
32nfcri 2895 . . . 4 𝑥 𝑧𝐴
4 dfssf.2 . . . . 5 𝑥𝐵
54nfcri 2895 . . . 4 𝑥 𝑧𝐵
63, 5nfim 1894 . . 3 𝑥(𝑧𝐴𝑧𝐵)
7 nfv 1912 . . 3 𝑧(𝑥𝐴𝑥𝐵)
8 eleq1w 2822 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
9 eleq1w 2822 . . . 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 1535  wcel 2106  wnfc 2888  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ss 3980
This theorem is referenced by:  dfss3f  3987  ssrd  4000  ssrmof  4063  ss2ab  4072  rankval4  9905  rabexgfGS  32527  ballotth  34519  dvcosre  45868  itgsinexplem1  45910
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