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Theorem dfss2f 3937
 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 2390. (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 3933 . 2 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 dfss2f.1 . . . . 5 𝑥𝐴
32nfcriv 2963 . . . 4 𝑥 𝑧𝐴
4 dfss2f.2 . . . . 5 𝑥𝐵
54nfcriv 2963 . . . 4 𝑥 𝑧𝐵
63, 5nfim 1897 . . 3 𝑥(𝑧𝐴𝑧𝐵)
7 nfv 1915 . . 3 𝑧(𝑥𝐴𝑥𝐵)
8 eleq1w 2893 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
9 eleq1w 2893 . . . 4 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
108, 9imbi12d 347 . . 3 (𝑧 = 𝑥 → ((𝑧𝐴𝑧𝐵) ↔ (𝑥𝐴𝑥𝐵)))
116, 7, 10cbvalv1 2361 . 2 (∀𝑧(𝑧𝐴𝑧𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
121, 11bitri 277 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   ∈ wcel 2114  Ⅎwnfc 2957   ⊆ wss 3913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-in 3920  df-ss 3930 This theorem is referenced by:  dfss3f  3938  ssrd  3951  ssrmof  4011  ss2ab  4018  rankval4  9274  rabexgfGS  30247  ballotth  31803  dvcosre  42345  itgsinexplem1  42387
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