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Theorem ss2ab 4037
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 2977 . . 3 𝑥{𝑥𝜑}
2 nfab1 2977 . . 3 𝑥{𝑥𝜓}
31, 2dfss2f 3956 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}))
4 abid 2801 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2801 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5imbi12i 353 . . 3 ((𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1813 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitri 277 1 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1528  wcel 2107  {cab 2797  wss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-in 3941  df-ss 3950
This theorem is referenced by:  abss  4038  ssab  4039  ss2abi  4041  ss2abdv  4042  ss2rab  4045  rabss2  4052  rabsssn  4599  clss2lem  39956  ssabf  41351  abssf  41363  sprssspr  43628
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