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Theorem ss2rabi 3103
Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
ss2rabi.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ss2rabi {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}

Proof of Theorem ss2rabi
StepHypRef Expression
1 ss2rab 3097 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
2 ss2rabi.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprgbir 2433 1 {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  {crab 2363  wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-in 3005  df-ss 3012
This theorem is referenced by:  supubti  6694  suplubti  6695
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