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Theorem ss2ab 3334
 Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
Assertion
Ref Expression
ss2ab ({x φ} {x ψ} ↔ x(φψ))

Proof of Theorem ss2ab
StepHypRef Expression
1 nfab1 2491 . . 3 x{x φ}
2 nfab1 2491 . . 3 x{x ψ}
31, 2dfss2f 3264 . 2 ({x φ} {x ψ} ↔ x(x {x φ} → x {x ψ}))
4 abid 2341 . . . 4 (x {x φ} ↔ φ)
5 abid 2341 . . . 4 (x {x ψ} ↔ ψ)
64, 5imbi12i 316 . . 3 ((x {x φ} → x {x ψ}) ↔ (φψ))
76albii 1566 . 2 (x(x {x φ} → x {x ψ}) ↔ x(φψ))
83, 7bitri 240 1 ({x φ} {x ψ} ↔ x(φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  {cab 2339   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  abss  3335  ssab  3336  ss2abi  3338  ss2abdv  3339  ss2rab  3342  rabss2  3349
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