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Mirrors > Home > ILE Home > Th. List > vtocl2 | GIF version |
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
vtocl2.1 | ⊢ 𝐴 ∈ V |
vtocl2.2 | ⊢ 𝐵 ∈ V |
vtocl2.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
vtocl2.4 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl2 | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocl2.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | 1 | isseti 2563 | . . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtocl2.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
4 | 3 | isseti 2563 | . . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵 |
5 | eeanv 1807 | . . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) | |
6 | vtocl2.3 | . . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
7 | 6 | biimpd 132 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓)) |
8 | 7 | 2eximi 1492 | . . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
9 | 5, 8 | sylbir 125 | . . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
10 | 2, 4, 9 | mp2an 402 | . . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓) |
11 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 11 | 19.36-1 1563 | . . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓)) |
13 | 10, 12 | eximii 1493 | . . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓) |
14 | 13 | 19.36aiv 1781 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓) |
15 | vtocl2.4 | . . 3 ⊢ 𝜑 | |
16 | 15 | ax-gen 1338 | . 2 ⊢ ∀𝑦𝜑 |
17 | 14, 16 | mpg 1340 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: caovord 5672 |
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