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Theorem bj-intabssel 9197
Description: Version of intss1 3621 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 2775 . . 3 x[A / x]φ
3 sbceq1a 2767 . . 3 (x = A → (φ[A / x]φ))
41, 2, 3elabgf 2679 . 2 (A 𝑉 → (A {xφ} ↔ [A / x]φ))
5 intss1 3621 . 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 1390  {cab 2023  wnfc 2162  [wsbc 2758  wss 2911   cint 3606
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by: (None)
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