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Theorem dfss2f 3094
 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.)
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 3092 . 2 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2 dfss2f.1 . . . . 5 𝑥𝐴
32nfcri 2276 . . . 4 𝑥 𝑧𝐴
4 dfss2f.2 . . . . 5 𝑥𝐵
54nfcri 2276 . . . 4 𝑥 𝑧𝐵
63, 5nfim 1552 . . 3 𝑥(𝑧𝐴𝑧𝐵)
7 nfv 1509 . . 3 𝑧(𝑥𝐴𝑥𝐵)
8 eleq1 2203 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
9 eleq1 2203 . . . 4 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
108, 9imbi12d 233 . . 3 (𝑧 = 𝑥 → ((𝑧𝐴𝑧𝐵) ↔ (𝑥𝐴𝑥𝐵)))
116, 7, 10cbval 1728 . 2 (∀𝑧(𝑧𝐴𝑧𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
121, 11bitri 183 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330   ∈ wcel 1481  Ⅎwnfc 2269   ⊆ wss 3077 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3083  df-ss 3090 This theorem is referenced by:  dfss3f  3095  ssrd  3108  ssrmof  3166  ss2ab  3171
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