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Mirrors > Home > MPE Home > Th. List > ralpr | Structured version Visualization version GIF version |
Description: Convert a 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 4181 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
6 | 1, 2, 5 | mp2an 704 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 {cpr 4127 |
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 df-pr 4128 |
This theorem is referenced by: fzprval 12271 fvinim0ffz 12449 wwlktovf1 13548 xpsfrnel 16046 xpsle 16064 isdrs2 16762 pmtrsn 17762 iblcnlem1 23360 wlkntrllem2 26090 wlkntrllem3 26091 2wlklem 26094 numclwwlkovf2ex 26613 subfacp1lem3 30418 fprb 30916 poimirlem1 32580 lfuhgr1v0e 40480 nbgr2vtx1edg 40572 nbuhgr2vtx1edgb 40574 umgr2v2evd2 40743 2Wlklem 40875 usgr2wlkneq 40962 usgr2trlncl 40966 21wlkdlem5 41136 21wlkdlem10 41142 3pthdlem1 41331 upgr4cycl4dv4e 41352 av-numclwwlkovf2ex 41517 ldepsnlinc 42091 |
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