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Mirrors > Home > MPE Home > Th. List > ralpr | Structured version Visualization version GIF version |
Description: Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralpr.1 | ⊢ 𝐴 ∈ V |
ralpr.2 | ⊢ 𝐵 ∈ V |
ralpr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralpr.4 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralpr | ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralpr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ralpr.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | ralpr.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | ralpr.4 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
5 | 3, 4 | ralprg 4660 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
6 | 1, 2, 5 | mp2an 691 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 Vcvv 3448 {cpr 4593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-v 3450 df-un 3920 df-sn 4592 df-pr 4594 |
This theorem is referenced by: fprb 7148 fzprval 13509 fvinim0ffz 13698 wwlktovf1 14853 xpsfrnel 17451 xpsle 17468 isdrs2 18202 pmtrsn 19308 iblcnlem1 25168 lfuhgr1v0e 28244 nbgr2vtx1edg 28340 nbuhgr2vtx1edgb 28342 umgr2v2evd2 28517 2wlklem 28657 2wlkdlem5 28916 2wlkdlem10 28922 clwwlknonex2lem2 29094 3pthdlem1 29150 upgr4cycl4dv4e 29171 subfacp1lem3 33816 poimirlem1 36108 paireqne 45777 requad2 45889 ldepsnlinc 46663 rrx2pnecoorneor 46875 rrx2line 46900 rrx2linest 46902 |
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