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Theorem dfss3f 3974
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 3973 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
4 df-ral 3063 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4bitr4i 278 1 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  wnfc 2884  wral 3062  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by:  nfss  3975  sigaclcu2  33149  bnj1498  34103  heibor1  36726  ssrabf  43851  ssrab2f  43854  limsupequzmpt2  44482  liminfequzmpt2  44555  pimconstlt1  45466  pimltpnff  45467  pimiooltgt  45474  pimdecfgtioc  45479  pimincfltioc  45480  pimdecfgtioo  45481  pimincfltioo  45482  pimgtmnff  45486  sssmf  45502
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