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Mirrors > Home > ILE Home > Th. List > spc2egv | GIF version |
Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
Ref | Expression |
---|---|
spc2egv.1 | ⊢ ((x = A ∧ y = B) → (φ ↔ ψ)) |
Ref | Expression |
---|---|
spc2egv | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (ψ → ∃x∃yφ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2562 | . . . 4 ⊢ (A ∈ 𝑉 → ∃x x = A) | |
2 | elisset 2562 | . . . 4 ⊢ (B ∈ 𝑊 → ∃y y = B) | |
3 | 1, 2 | anim12i 321 | . . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (∃x x = A ∧ ∃y y = B)) |
4 | eeanv 1804 | . . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B)) | |
5 | 3, 4 | sylibr 137 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∃x∃y(x = A ∧ y = B)) |
6 | spc2egv.1 | . . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ)) | |
7 | 6 | biimprcd 149 | . . 3 ⊢ (ψ → ((x = A ∧ y = B) → φ)) |
8 | 7 | 2eximdv 1759 | . 2 ⊢ (ψ → (∃x∃y(x = A ∧ y = B) → ∃x∃yφ)) |
9 | 5, 8 | syl5com 26 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (ψ → ∃x∃yφ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-v 2553 |
This theorem is referenced by: spc2ev 2642 th3q 6147 addnnnq0 6432 mulnnnq0 6433 addsrpr 6673 mulsrpr 6674 |
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