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Theorem ss2abdv 3083
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
Hypothesis
Ref Expression
ss2abdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ss2abdv (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem ss2abdv
StepHypRef Expression
1 ss2abdv.1 . . 3 (𝜑 → (𝜓𝜒))
21alrimiv 1799 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 ss2ab 3078 . 2 ({𝑥𝜓} ⊆ {𝑥𝜒} ↔ ∀𝑥(𝜓𝜒))
42, 3sylibr 132 1 (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  {cab 2071  wss 2988
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-in 2994  df-ss 3001
This theorem is referenced by:  ssopab2  4074  iotass  4959  imadif  5055  imain  5057  opabbrex  5643  ssoprab2  5655  tfr1onlemssrecs  6051  tfrcllemssrecs  6064
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