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Theorem dfss2f 3812
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 2334. (Revised by Gino Giotto, 19-May-2023.)
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 3809 . 2 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 dfss2f.1 . . . . 5 𝑥𝐴
32nfcriv 2925 . . . 4 𝑥 𝑧𝐴
4 dfss2f.2 . . . . 5 𝑥𝐵
54nfcriv 2925 . . . 4 𝑥 𝑧𝐵
63, 5nfim 1943 . . 3 𝑥(𝑧𝐴𝑧𝐵)
7 nfv 1957 . . 3 𝑧(𝑥𝐴𝑥𝐵)
8 eleq1w 2842 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
9 eleq1w 2842 . . . 4 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
108, 9imbi12d 336 . . 3 (𝑧 = 𝑥 → ((𝑧𝐴𝑧𝐵) ↔ (𝑥𝐴𝑥𝐵)))
116, 7, 10cbvalv1 2312 . 2 (∀𝑧(𝑧𝐴𝑧𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
121, 11bitri 267 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1599  wcel 2107  wnfc 2919  wss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-in 3799  df-ss 3806
This theorem is referenced by:  dfss3f  3813  ssrd  3826  ss2ab  3891  rankval4  9029  ssrmo  29917  rabexgfGS  29919  ballotth  31206  dvcosre  41068  itgsinexplem1  41111
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