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Mirrors > Home > NFE Home > Th. List > ralimi2 | GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
ralimi2.1 | ⊢ ((x ∈ A → φ) → (x ∈ B → ψ)) |
Ref | Expression |
---|---|
ralimi2 | ⊢ (∀x ∈ A φ → ∀x ∈ B ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimi2.1 | . . 3 ⊢ ((x ∈ A → φ) → (x ∈ B → ψ)) | |
2 | 1 | alimi 1559 | . 2 ⊢ (∀x(x ∈ A → φ) → ∀x(x ∈ B → ψ)) |
3 | df-ral 2619 | . 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ)) | |
4 | df-ral 2619 | . 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ)) | |
5 | 2, 3, 4 | 3imtr4i 257 | 1 ⊢ (∀x ∈ A φ → ∀x ∈ B ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-ral 2619 |
This theorem is referenced by: ralimia 2687 ralcom3 2776 peano5 4409 |
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