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Mirrors > Home > ILE Home > Th. List > sbciedf | GIF version |
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Ref | Expression |
---|---|
sbcied.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sbcied.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
sbciedf.3 | ⊢ Ⅎ𝑥𝜑 |
sbciedf.4 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
Ref | Expression |
---|---|
sbciedf | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcied.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sbciedf.4 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
3 | sbciedf.3 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | sbcied.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
5 | 4 | ex 108 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
6 | 3, 5 | alrimi 1415 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
7 | sbciegft 2793 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | |
8 | 1, 2, 6, 7 | syl3anc 1135 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 Ⅎwnf 1349 ∈ 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-3an 887 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: sbcied 2799 sbc2iegf 2828 csbiebt 2886 sbcnestgf 2897 ovmpt2dxf 5626 |
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