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Mirrors > Home > MPE Home > Th. List > ralprg | Structured version Visualization version GIF version |
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralprg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralprg.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4371 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | raleqi 3325 | . . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑) |
3 | ralunb 3992 | . . 3 ⊢ (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)) | |
4 | 2, 3 | bitri 267 | . 2 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑)) |
5 | ralprg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 5 | ralsng 4409 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
7 | ralprg.2 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
8 | 7 | ralsng 4409 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜒)) |
9 | 6, 8 | bi2anan9 630 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∧ 𝜒))) |
10 | 4, 9 | syl5bb 275 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∪ cun 3767 {csn 4368 {cpr 4370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-v 3387 df-sbc 3634 df-un 3774 df-sn 4369 df-pr 4371 |
This theorem is referenced by: raltpg 4426 ralpr 4428 iinxprg 4791 disjprg 4839 fpropnf1 6752 f12dfv 6757 f13dfv 6758 suppr 8619 infpr 8651 pfx2 14032 sumpr 14818 gcdcllem2 15557 lcmfpr 15675 joinval2lem 17323 meetval2lem 17337 sgrp2rid2 17729 sgrp2nmndlem4 17731 sgrp2nmndlem5 17732 iccntr 22952 limcun 24000 cplgr3v 26685 3wlkdlem4 27506 frgr3v 27624 3vfriswmgr 27627 prsiga 30710 |
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