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Mirrors > Home > MPE Home > Th. List > vtocl2 | Structured version Visualization version 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 3349 | . . . . 5 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtocl2.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
4 | 3 | isseti 3349 | . . . . 5 ⊢ ∃𝑦 𝑦 = 𝐵 |
5 | eeanv 2327 | . . . . . 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 219 | . . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 → 𝜓)) |
8 | 7 | 2eximi 1912 | . . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
9 | 5, 8 | sylbir 225 | . . . . 5 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥∃𝑦(𝜑 → 𝜓)) |
10 | 2, 4, 9 | mp2an 710 | . . . 4 ⊢ ∃𝑥∃𝑦(𝜑 → 𝜓) |
11 | 19.36v 2069 | . . . . 5 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦𝜑 → 𝜓)) | |
12 | 11 | exbii 1923 | . . . 4 ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ ∃𝑥(∀𝑦𝜑 → 𝜓)) |
13 | 10, 12 | mpbi 220 | . . 3 ⊢ ∃𝑥(∀𝑦𝜑 → 𝜓) |
14 | 13 | 19.36iv 2023 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓) |
15 | vtocl2.4 | . . 3 ⊢ 𝜑 | |
16 | 15 | ax-gen 1871 | . 2 ⊢ ∀𝑦𝜑 |
17 | 14, 16 | mpg 1873 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1630 = wceq 1632 ∃wex 1853 ∈ wcel 2139 Vcvv 3340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-v 3342 |
This theorem is referenced by: caovord 7010 sornom 9291 wloglei 10752 ipodrsima 17366 mpfind 19738 mclsppslem 31787 monotoddzzfi 38009 |
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