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Mirrors > Home > ILE Home > Th. List > sbcimdv | GIF version |
Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
sbcimdv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
sbcimdv | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcimdv.1 | . . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alrimiv 1754 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | spsbc 2775 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
4 | 2, 3 | syl5 28 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) |
5 | sbcimg 2804 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) | |
6 | 4, 5 | sylibd 138 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
7 | 6 | impcom 116 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∈ wcel 1393 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sbc 2765 |
This theorem is referenced by: (None) |
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