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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 4596 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
6 | 1, 2, 5 | mp2an 692 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 {cpr 4529 |
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 2018 ax-8 2114 ax-9 2122 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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-v 3400 df-un 3858 df-sn 4528 df-pr 4530 |
This theorem is referenced by: fprb 6987 fzprval 13138 fvinim0ffz 13326 wwlktovf1 14489 xpsfrnel 17021 xpsle 17038 isdrs2 17767 pmtrsn 18865 iblcnlem1 24639 lfuhgr1v0e 27296 nbgr2vtx1edg 27392 nbuhgr2vtx1edgb 27394 umgr2v2evd2 27569 2wlklem 27709 2wlkdlem5 27967 2wlkdlem10 27973 clwwlknonex2lem2 28145 3pthdlem1 28201 upgr4cycl4dv4e 28222 subfacp1lem3 32811 poimirlem1 35464 paireqne 44579 requad2 44691 ldepsnlinc 45465 rrx2pnecoorneor 45677 rrx2line 45702 rrx2linest 45704 |
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