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Theorem ss2abdv 3007
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 1751 . 2 (φx(ψχ))
3 ss2ab 3002 . 2 ({xψ} ⊆ {xχ} ↔ x(ψχ))
42, 3sylibr 137 1 (φ → {xψ} ⊆ {xχ})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  {cab 2023  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925
This theorem is referenced by:  ssopab2  4003  iotass  4827  imadif  4922  imain  4924  opabbrex  5491  ssoprab2  5503
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