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Theorem dfss3f 3628
Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
Hypotheses
Ref Expression
dfss2f.1 𝑥𝐴
dfss2f.2 𝑥𝐵
Assertion
Ref Expression
dfss3f (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)

Proof of Theorem dfss3f
StepHypRef Expression
1 dfss2f.1 . . 3 𝑥𝐴
2 dfss2f.2 . . 3 𝑥𝐵
31, 2dfss2f 3627 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
4 df-ral 2946 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4bitr4i 267 1 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wcel 2030  wnfc 2780  wral 2941  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-in 3614  df-ss 3621
This theorem is referenced by:  nfss  3629  sigaclcu2  30311  bnj1498  31255  heibor1  33739  ssrabf  39612  ssrab2f  39614  limsupequzmpt2  40268  liminfequzmpt2  40341  pimconstlt1  41236  pimltpnf  41237  pimiooltgt  41242  pimdecfgtioc  41246  pimincfltioc  41247  pimdecfgtioo  41248  pimincfltioo  41249  sssmf  41268
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