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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 3507 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | elisset 3507 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
3 | 1, 2 | anim12i 614 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) |
4 | exdistrv 1956 | . . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) | |
5 | 3, 4 | sylibr 236 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
6 | spc2egv.1 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
7 | 6 | biimprcd 252 | . . 3 ⊢ (𝜓 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜑)) |
8 | 7 | 2eximdv 1920 | . 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 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 |
This theorem is referenced by: spc2gv 3603 spc3egv 3606 spc2ev 3610 tpres 6965 addsrpr 10499 mulsrpr 10500 2pthon3v 27724 umgr2wlk 27730 0pthonv 27910 1pthon2v 27934 satfv1 32612 sat1el2xp 32628 dvnprodlem1 42238 dfatcolem 43461 fundcmpsurbijinj 43577 |
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