ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss2abi GIF version

Theorem ss2abi 3200
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 3196 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
2 ss2abi.1 . 2 (𝜑𝜓)
31, 2mpgbir 1433 1 {𝑥𝜑} ⊆ {𝑥𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  {cab 2143  wss 3102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115
This theorem is referenced by:  abssi  3203  rabssab  3216  pwsnss  3768  iinuniss  3933  pwpwssunieq  3939  abssexg  4145  imassrn  4941  imadiflem  5251  imainlem  5253  fabexg  5359  f1oabexg  5428  tfrcllemssrecs  6301  mapex  6601  tgval  12519  tgvalex  12520
  Copyright terms: Public domain W3C validator