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Mirrors > Home > MPE Home > Th. List > alral | Structured version Visualization version GIF version |
Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.) |
Ref | Expression |
---|---|
alral | ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ala1 1815 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∈ wcel 2106 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 206 df-ral 3062 |
This theorem is referenced by: abnex 7740 find 7883 findOLD 7884 brdom5 10520 brdom4 10521 hashgt23el 14380 prodeq2w 15852 rpnnen2lem12 16164 umgr2cycllem 34119 umgr2cycl 34120 elpotr 34741 fvineqsnf1 36279 fvineqsneq 36281 phpreu 36460 ordelordALTVD 43613 rexrsb 45794 |
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