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Theorem ss2abi 3012
 Description: Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
Hypothesis
Ref Expression
ss2abi.1 (𝜑𝜓)
Assertion
Ref Expression
ss2abi {𝑥𝜑} ⊆ {𝑥𝜓}

Proof of Theorem ss2abi
StepHypRef Expression
1 ss2ab 3008 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
2 ss2abi.1 . 2 (𝜑𝜓)
31, 2mpgbir 1342 1 {𝑥𝜑} ⊆ {𝑥𝜓}
 Colors of variables: wff set class Syntax hints:   → wi 4  {cab 2026   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931 This theorem is referenced by:  abssi  3015  rabssab  3027  pwsnss  3574  iinuniss  3737  abssexg  3934  imassrn  4679  imadiflem  4978  imainlem  4980  fabexg  5077  f1oabexg  5138
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