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Theorem ss2rabi 3252
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 3246 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
2 ss2rabi.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprgbir 2548 1 {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  {crab 2472  wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-in 3150  df-ss 3157
This theorem is referenced by:  supubti  7029  suplubti  7030
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