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Mirrors > Home > ILE 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 1341 | . 2 ⊢ (∀x(x ∈ A → φ) → ∀x(x ∈ B → ψ)) |
3 | df-ral 2305 | . 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ)) | |
4 | df-ral 2305 | . 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ)) | |
5 | 2, 3, 4 | 3imtr4i 190 | 1 ⊢ (∀x ∈ A φ → ∀x ∈ B ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 ∈ wcel 1390 ∀wral 2300 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 |
This theorem depends on definitions: df-bi 110 df-ral 2305 |
This theorem is referenced by: ralimia 2376 ralcom3 2471 bj-nntrans 9411 bj-findis 9439 |
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