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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 377 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) |
7 | 4, 6 | sbcied 2945 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
8 | 2, 7 | sbcied 2945 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 [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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 |
This theorem is referenced by: dfoprab3 6089 |
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