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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intabssel | GIF version |
Description: Version of intss1 3651 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 2832 | . . 3 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 |
3 | sbceq1a 2824 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | elabgf 2736 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)) |
5 | intss1 3651 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) | |
6 | 4, 5 | syl6bir 162 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 {cab 2067 Ⅎwnfc 2206 [wsbc 2815 ⊆ wss 2973 ∩ cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-in 2979 df-ss 2986 df-int 3637 |
This theorem is referenced by: (None) |
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