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Theorem ss2abdv 3165
 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 1846 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 ss2ab 3160 . 2 ({𝑥𝜓} ⊆ {𝑥𝜒} ↔ ∀𝑥(𝜓𝜒))
42, 3sylibr 133 1 (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329  {cab 2123   ⊆ wss 3066 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079 This theorem is referenced by:  ssopab2  4192  iotass  5100  imadif  5198  imain  5200  opabbrex  5808  ssoprab2  5820  tfr1onlemssrecs  6229  tfrcllemssrecs  6242  ss2ixp  6598
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