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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 2760 | . . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtocl2.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
4 | 3 | isseti 2760 | . . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵 |
5 | eeanv 1944 | . . . . . 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 144 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓)) |
8 | 7 | 2eximi 1612 | . . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
9 | 5, 8 | sylbir 135 | . . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
10 | 2, 4, 9 | mp2an 426 | . . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓) |
11 | nfv 1539 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 11 | 19.36-1 1684 | . . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓)) |
13 | 10, 12 | eximii 1613 | . . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓) |
14 | 13 | 19.36aiv 1913 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓) |
15 | vtocl2.4 | . . 3 ⊢ 𝜑 | |
16 | 15 | ax-gen 1460 | . 2 ⊢ ∀𝑦𝜑 |
17 | 14, 16 | mpg 1462 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-v 2754 |
This theorem is referenced by: undifexmid 4211 caovord 6069 ctssexmid 7179 |
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