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Mirrors > Home > ILE Home > Th. List > sbc2iedv | GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
Ref | Expression |
---|---|
sbc2iedv.1 | ⊢ 𝐴 ∈ V |
sbc2iedv.2 | ⊢ 𝐵 ∈ V |
sbc2iedv.3 | ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
sbc2iedv | ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc2iedv.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | sbc2iedv.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V) |
5 | sbc2iedv.3 | . . . 4 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) | |
6 | 5 | impl 362 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) |
7 | 4, 6 | sbcied 2799 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
8 | 2, 7 | sbcied 2799 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 [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: dfoprab3 5817 |
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