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Mirrors > Home > ILE Home > Th. List > sbcthdv | GIF version |
Description: Deduction version of sbcth 2922. (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 1846 | . 2 ⊢ (𝜑 → ∀𝑥𝜓) |
3 | spsbc 2920 | . 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 1329 ∈ wcel 1480 [wsbc 2909 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 df-sbc 2910 |
This theorem is referenced by: (None) |
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