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Mirrors > Home > NFE Home > Th. List > ralim | GIF version |
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.) |
Ref | Expression |
---|---|
ralim | ⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A φ → ∀x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2619 | . . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ))) | |
2 | ax-2 7 | . . . 4 ⊢ ((x ∈ A → (φ → ψ)) → ((x ∈ A → φ) → (x ∈ A → ψ))) | |
3 | 2 | al2imi 1561 | . . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) → (∀x(x ∈ A → φ) → ∀x(x ∈ A → ψ))) |
4 | 1, 3 | sylbi 187 | . 2 ⊢ (∀x ∈ A (φ → ψ) → (∀x(x ∈ A → φ) → ∀x(x ∈ A → ψ))) |
5 | df-ral 2619 | . 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ)) | |
6 | df-ral 2619 | . 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ)) | |
7 | 4, 5, 6 | 3imtr4g 261 | 1 ⊢ (∀x ∈ A (φ → ψ) → (∀x ∈ A φ → ∀x ∈ A ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-ral 2619 |
This theorem is referenced by: ral2imi 2690 r19.30 2756 |
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