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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 4105 | . 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)) | |
2 | ralunsn.1 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ralsng 4589 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜓)) |
4 | 3 | anbi2d 632 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
5 | 1, 4 | syl5bb 286 | 1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∪ cun 3864 {csn 4541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-v 3410 df-un 3871 df-sn 4542 |
This theorem is referenced by: 2ralunsn 4806 symgextfo 18814 gsmsymgrfixlem1 18819 gsmsymgreqlem2 18823 symgfixf1 18829 cply1coe0bi 21221 scmatf1 21428 mdetunilem9 21517 m2cpminvid2lem 21651 tgcgr4 26622 clwlkclwwlklem2a1 28075 clwlkclwwlkf1lem3 28089 disjunsn 30652 |
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