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Mirrors > Home > MPE Home > Th. List > spc2gv | Structured version Visualization version GIF version |
Description: Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.) |
Ref | Expression |
---|---|
spc2egv.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spc2gv | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spc2egv.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 318 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | spc2egv 3560 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ∃𝑥∃𝑦 ¬ 𝜑)) |
4 | 2nalexn 1831 | . . 3 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) | |
5 | 3, 4 | syl6ibr 252 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ¬ ∀𝑥∀𝑦𝜑)) |
6 | 5 | con4d 115 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-clel 2811 |
This theorem is referenced by: rspc2gv 3591 trel 5235 elovmpo 7602 seqf1olem2 13957 seqf1o 13958 fi1uzind 14405 brfi1indALT 14408 pslem 18469 cnmpt12 23041 cnmpt22 23048 mclsppslem 34241 mbfresfi 36174 lpolconN 40000 ismrcd2 41069 ismrc 41071 |
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