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Mirrors > Home > MPE Home > Th. List > spc2gv | Structured version Visualization version GIF version |
Description: Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.) |
Ref | Expression |
---|---|
spc2egv.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spc2gv | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spc2egv.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 307 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | spc2egv 3399 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ∃𝑥∃𝑦 ¬ 𝜑)) |
4 | 2nalexn 1868 | . . 3 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) | |
5 | 3, 4 | syl6ibr 242 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ¬ ∀𝑥∀𝑦𝜑)) |
6 | 5 | con4d 114 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1594 = wceq 1596 ∃wex 1817 ∈ wcel 2103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-ext 2704 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-clab 2711 df-cleq 2717 df-clel 2720 df-v 3306 |
This theorem is referenced by: rspc2gv 3425 trel 4867 elovmpt2 6996 seqf1olem2 12956 seqf1o 12957 fi1uzind 13392 brfi1indALT 13395 pslem 17328 cnmpt12 21593 cnmpt22 21600 mclsppslem 31708 mbfresfi 33688 lpolconN 37195 ismrcd2 37681 ismrc 37683 |
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