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Theorem dfss3f 3059
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 3058 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
4 df-ral 2398 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
53, 4bitr4i 186 1 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314  wcel 1465  wnfc 2245  wral 2393  wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-in 3047  df-ss 3054
This theorem is referenced by:  nfss  3060
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