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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 1390 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) |
3 | df-ral 2365 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | df-ral 2365 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
5 | 2, 3, 4 | 3imtr4i 200 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1288 ∈ wcel 1439 ∀wral 2360 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 |
This theorem depends on definitions: df-bi 116 df-ral 2365 |
This theorem is referenced by: ralimia 2437 ralcom3 2537 bj-nntrans 12150 bj-findis 12178 |
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