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Mirrors > Home > ILE Home > Th. List > ralimiaa | 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 115 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
3 | 2 | ralimia 2548 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2158 ∀wral 2465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1457 ax-gen 1459 |
This theorem depends on definitions: df-bi 117 df-ral 2470 |
This theorem is referenced by: ralrnmpt 5671 rexrnmpt 5672 acexmidlem2 5885 mptelixpg 6748 trirec0 15146 |
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