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Theorem ss2abdv 3220
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 1867 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 ss2ab 3215 . 2 ({𝑥𝜓} ⊆ {𝑥𝜒} ↔ ∀𝑥(𝜓𝜒))
42, 3sylibr 133 1 (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1346  {cab 2156  wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134
This theorem is referenced by:  ssopab2  4260  iotass  5177  imadif  5278  imain  5280  opabbrex  5897  ssoprab2  5909  tfr1onlemssrecs  6318  tfrcllemssrecs  6331  ss2ixp  6689
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