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Mirrors > Home > ILE Home > Th. List > sbcthdv | GIF version |
Description: Deduction version of sbcth 2895. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
sbcthdv.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
sbcthdv | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcthdv.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | alrimiv 1830 | . 2 ⊢ (𝜑 → ∀𝑥𝜓) |
3 | spsbc 2893 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓)) | |
4 | 2, 3 | mpan9 279 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1314 ∈ wcel 1465 [wsbc 2882 |
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-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-v 2662 df-sbc 2883 |
This theorem is referenced by: (None) |
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