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| Mirrors > Home > MPE Home > Th. List > spc2egv | Structured version Visualization version GIF version | ||
| Description: Existential specialization with two 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 2823 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
| 2 | elisset 2823 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
| 3 | 1, 2 | anim12i 613 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) |
| 4 | exdistrv 1955 | . . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) | |
| 5 | 3, 4 | sylibr 234 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
| 6 | spc2egv.1 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
| 7 | 6 | biimprcd 250 | . . 3 ⊢ (𝜓 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜑)) |
| 8 | 7 | 2eximdv 1919 | . 2 ⊢ (𝜓 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜑)) |
| 9 | 5, 8 | syl5com 31 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-clel 2816 |
| This theorem is referenced by: spc2gv 3600 spc3egv 3603 spc2ev 3607 tpres 7221 addsrpr 11115 mulsrpr 11116 2pthon3v 29963 umgr2wlk 29969 0pthonv 30148 1pthon2v 30172 satfv1 35368 sat1el2xp 35384 dvnprodlem1 45961 dfatcolem 47267 fundcmpsurbijinj 47397 |
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