Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-intabssel Structured version   GIF version

Theorem bj-intabssel 7035
 Description: Version of intss1 3604 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
Hypothesis
Ref Expression
bj-intabssel.nf xA
Assertion
Ref Expression
bj-intabssel (A 𝑉 → ([A / x]φ {xφ} ⊆ A))

Proof of Theorem bj-intabssel
StepHypRef Expression
1 bj-intabssel.nf . . 3 xA
21nfsbc1 2758 . . 3 x[A / x]φ
3 sbceq1a 2750 . . 3 (x = A → (φ[A / x]φ))
41, 2, 3elabgf 2662 . 2 (A 𝑉 → (A {xφ} ↔ [A / x]φ))
5 intss1 3604 . 2 (A {xφ} → {xφ} ⊆ A)
64, 5syl6bir 153 1 (A 𝑉 → ([A / x]φ {xφ} ⊆ A))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1374  {cab 2008  Ⅎwnfc 2147  [wsbc 2741   ⊆ wss 2894  ∩ cint 3589 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-in 2901  df-ss 2908  df-int 3590 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator