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Theorem ss2abi 3137
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 3133 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
2 ss2abi.1 . 2 (𝜑𝜓)
31, 2mpgbir 1412 1 {𝑥𝜑} ⊆ {𝑥𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  {cab 2101  wss 3039
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-in 3045  df-ss 3052
This theorem is referenced by:  abssi  3140  rabssab  3152  pwsnss  3698  iinuniss  3863  pwpwssunieq  3869  abssexg  4074  imassrn  4860  imadiflem  5170  imainlem  5172  fabexg  5278  f1oabexg  5345  tfrcllemssrecs  6215  mapex  6514  tgval  12124  tgvalex  12125
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