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Mirrors > Home > NFE 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 2869 | . . . 4 ⊢ (A ∈ V → ∃x x = A) | |
2 | elisset 2869 | . . . 4 ⊢ (B ∈ W → ∃y y = B) | |
3 | 1, 2 | anim12i 549 | . . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (∃x x = A ∧ ∃y y = B)) |
4 | eeanv 1913 | . . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B)) | |
5 | 3, 4 | sylibr 203 | . 2 ⊢ ((A ∈ V ∧ B ∈ W) → ∃x∃y(x = A ∧ y = B)) |
6 | spc2egv.1 | . . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ)) | |
7 | 6 | biimprcd 216 | . . 3 ⊢ (ψ → ((x = A ∧ y = B) → φ)) |
8 | 7 | 2eximdv 1624 | . 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 setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-v 2861 |
This theorem is referenced by: spc2gv 2942 spc2ev 2947 |
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