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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 2668 | . . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtocl2.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
4 | 3 | isseti 2668 | . . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵 |
5 | eeanv 1884 | . . . . . 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 143 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓)) |
8 | 7 | 2eximi 1565 | . . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
9 | 5, 8 | sylbir 134 | . . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
10 | 2, 4, 9 | mp2an 422 | . . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓) |
11 | nfv 1493 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 11 | 19.36-1 1636 | . . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓)) |
13 | 10, 12 | eximii 1566 | . . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓) |
14 | 13 | 19.36aiv 1857 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓) |
15 | vtocl2.4 | . . 3 ⊢ 𝜑 | |
16 | 15 | ax-gen 1410 | . 2 ⊢ ∀𝑦𝜑 |
17 | 14, 16 | mpg 1412 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1314 = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 |
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-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-v 2662 |
This theorem is referenced by: undifexmid 4087 caovord 5910 ctssexmid 6992 |
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