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Mirrors > Home > MPE Home > Th. List > ralunsn | Structured version Visualization version GIF version |
Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralunsn.1 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ralunsn | ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralunb 3756 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)) | |
2 | ralunsn.1 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ralsng 4165 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜓)) |
4 | 3 | anbi2d 736 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
5 | 1, 4 | syl5bb 271 | 1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ cun 3538 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-sbc 3403 df-un 3545 df-sn 4126 |
This theorem is referenced by: 2ralunsn 4361 symgextfo 17665 gsmsymgrfixlem1 17670 gsmsymgreqlem2 17674 symgfixf1 17680 cply1coe0bi 19491 scmatf1 20156 mdetunilem9 20245 m2cpminvid2lem 20378 tgcgr4 25226 clwlkisclwwlklem2a1 26307 clwlkf1clwwlklem 26376 disjunsn 28789 clwlkclwwlklem2a1 41201 clwlksf1clwwlklem 41275 |
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