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Theorem ss2rabi 3022
 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 3016 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
2 ss2rabi.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprgbir 2379 1 {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  {crab 2310   ⊆ 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-ral 2311  df-rab 2315  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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