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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 113 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
6 | 3, 5 | alrimi 1455 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
7 | sbciegft 2844 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | |
8 | 1, 2, 6, 7 | syl3anc 1169 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 [wsbc 2815 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 |
This theorem is referenced by: sbcied 2850 sbc2iegf 2884 csbiebt 2942 sbcnestgf 2953 ovmpt2dxf 5646 |
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