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Theorem ss2ab 4017
Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
Assertion
Ref Expression
ss2ab ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))

Proof of Theorem ss2ab
StepHypRef Expression
1 nfab1 2906 . . 3 𝑥{𝑥𝜑}
2 nfab1 2906 . . 3 𝑥{𝑥𝜓}
31, 2dfss2f 3935 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}))
4 abid 2714 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2714 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5imbi12i 351 . . 3 ((𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1822 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitri 275 1 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  {cab 2710  wss 3911
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  abss  4018  ssab  4019  ss2abdvOLD  4023  ss2abiOLD  4025  ss2rab  4029  rabss2  4036  rabsssn  4629  bj-gabss  35451  clss2lem  41971  ssabf  43398  abssf  43410  cfsetssfset  45376  sprssspr  45759
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