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Theorem bj-intabssel 14423
Description: Version of intss1 3859 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 2980 . . 3 𝑥[𝐴 / 𝑥]𝜑
3 sbceq1a 2972 . . 3 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
41, 2, 3elabgf 2879 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑))
5 intss1 3859 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6bir 164 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 {𝑥𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  {cab 2163  wnfc 2306  [wsbc 2962  wss 3129   cint 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-in 3135  df-ss 3142  df-int 3845
This theorem is referenced by: (None)
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