![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ralimiaa | Structured version Visualization version GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.) |
Ref | Expression |
---|---|
ralimiaa.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
Ref | Expression |
---|---|
ralimiaa | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimiaa.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) | |
2 | 1 | ex 412 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
3 | 2 | ralimia 3078 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ral 3060 |
This theorem is referenced by: ralrnmptw 7114 ralrnmpt 7116 tz7.48-2 8481 mptelixpg 8974 boxriin 8979 acnlem 10086 iundom2g 10578 konigthlem 10606 hashge2el2dif 14516 rlim2 15529 prdsbas3 17528 prdsdsval2 17531 ptbasfi 23605 ptunimpt 23619 voliun 25603 lgamgulmlem6 27092 riesz4i 32092 dmdbr6ati 32452 ctbssinf 37389 |
Copyright terms: Public domain | W3C validator |