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Mirrors > Home > MPE Home > Th. List > sbc2iedv | Structured version Visualization version 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 11 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | sbc2iedv.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V) |
5 | sbc2iedv.3 | . . . 4 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) | |
6 | 5 | impl 648 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)) |
7 | 4, 6 | sbcied 3439 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
8 | 2, 7 | sbcied 3439 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 |
This theorem is referenced by: dfoprab3 7115 sbcie2s 15744 ismnddef 17119 sdclem1 32709 |
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