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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intabssel | GIF version |
Description: Version of intss1 3794 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) |
Ref | Expression |
---|---|
bj-intabssel.nf | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
bj-intabssel | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-intabssel.nf | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfsbc1 2930 | . . 3 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 |
3 | sbceq1a 2922 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | elabgf 2830 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)) |
5 | intss1 3794 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) | |
6 | 4, 5 | syl6bir 163 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 {cab 2126 Ⅎwnfc 2269 [wsbc 2913 ⊆ wss 3076 ∩ cint 3779 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-sbc 2914 df-in 3082 df-ss 3089 df-int 3780 |
This theorem is referenced by: (None) |
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