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Mirrors > Home > MPE Home > Th. List > spc2d | Structured version Visualization version GIF version |
Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
Ref | Expression |
---|---|
spc2ed.x | ⊢ Ⅎ𝑥𝜒 |
spc2ed.y | ⊢ Ⅎ𝑦𝜒 |
spc2ed.1 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
spc2d | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∀𝑥∀𝑦𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nalexn 1834 | . . 3 ⊢ (¬ ∀𝑥∀𝑦𝜓 ↔ ∃𝑥∃𝑦 ¬ 𝜓) | |
2 | 1 | con1bii 357 | . 2 ⊢ (¬ ∃𝑥∃𝑦 ¬ 𝜓 ↔ ∀𝑥∀𝑦𝜓) |
3 | spc2ed.x | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
4 | 3 | nfn 1864 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜒 |
5 | spc2ed.y | . . . . 5 ⊢ Ⅎ𝑦𝜒 | |
6 | 5 | nfn 1864 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜒 |
7 | spc2ed.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
8 | 7 | notbid 318 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (¬ 𝜓 ↔ ¬ 𝜒)) |
9 | 4, 6, 8 | spc2ed 3539 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ 𝜒 → ∃𝑥∃𝑦 ¬ 𝜓)) |
10 | 9 | con1d 145 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ ∃𝑥∃𝑦 ¬ 𝜓 → 𝜒)) |
11 | 2, 10 | syl5bir 242 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∀𝑥∀𝑦𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1540 = wceq 1542 ∃wex 1786 Ⅎwnf 1790 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-clel 2818 |
This theorem is referenced by: (None) |
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