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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 1777 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | df-ral 3086 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
3 | 1, 2 | sylibr 226 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1506 ∈ wcel 2051 ∀wral 3081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 |
This theorem depends on definitions: df-bi 199 df-ral 3086 |
This theorem is referenced by: abnex 7294 find 7420 brdom5 9747 brdom4 9748 hashgt23el 13596 prodeq2w 15124 rpnnen2lem12 15436 elpotr 32583 fvineqsnf1 34169 fvineqsneq 34171 phpreu 34354 neik0pk1imk0 39798 ordelordALTVD 40658 rexrsb 42738 |
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