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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intabssel | GIF version |
Description: Version of intss1 3859 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 2980 | . . 3 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 |
3 | sbceq1a 2972 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | elabgf 2879 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)) |
5 | intss1 3859 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴) | |
6 | 4, 5 | syl6bir 164 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 {cab 2163 Ⅎwnfc 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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