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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 4632 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
6 | 1, 2, 5 | mp2an 690 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 {cpr 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-sbc 3773 df-un 3941 df-sn 4568 df-pr 4570 |
This theorem is referenced by: fprb 6956 fzprval 12969 fvinim0ffz 13157 wwlktovf1 14321 xpsfrnel 16835 xpsle 16852 isdrs2 17549 pmtrsn 18647 iblcnlem1 24388 lfuhgr1v0e 27036 nbgr2vtx1edg 27132 nbuhgr2vtx1edgb 27134 umgr2v2evd2 27309 2wlklem 27449 2wlkdlem5 27708 2wlkdlem10 27714 clwwlknonex2lem2 27887 3pthdlem1 27943 upgr4cycl4dv4e 27964 subfacp1lem3 32429 poimirlem1 34908 paireqne 43693 requad2 43808 ldepsnlinc 44583 rrx2pnecoorneor 44722 rrx2line 44747 rrx2linest 44749 |
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