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Theorem bj-intabssel 12807
Description: Version of intss1 3754 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 2897 . . 3 𝑥[𝐴 / 𝑥]𝜑
3 sbceq1a 2889 . . 3 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
41, 2, 3elabgf 2798 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑))
5 intss1 3754 . 2 (𝐴 ∈ {𝑥𝜑} → {𝑥𝜑} ⊆ 𝐴)
64, 5syl6bir 163 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 {𝑥𝜑} ⊆ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  {cab 2101  wnfc 2243  [wsbc 2880  wss 3039   cint 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-in 3045  df-ss 3052  df-int 3740
This theorem is referenced by: (None)
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