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Theorem bj-intabssel 9928
 Description: Version of intss1 3630 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 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 {𝑥𝜑} ⊆ 𝐴))

Proof of Theorem bj-intabssel
StepHypRef Expression
1 bj-intabssel.nf . . 3 𝑥𝐴
21nfsbc1 2781 . . 3 𝑥[𝐴 / 𝑥]𝜑
3 sbceq1a 2773 . . 3 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
41, 2, 3elabgf 2685 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑))
5 intss1 3630 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6bir 153 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 {𝑥𝜑} ⊆ 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  [wsbc 2764   ⊆ wss 2917  ∩ cint 3615 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-in 2924  df-ss 2931  df-int 3616 This theorem is referenced by: (None)
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