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Mirrors > Home > ILE 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 2353 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
2 | ax-2 6 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜓))) | |
3 | 2 | al2imi 1387 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
4 | 1, 3 | sylbi 119 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) |
5 | df-ral 2353 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
6 | df-ral 2353 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
7 | 4, 5, 6 | 3imtr4g 203 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 ∈ wcel 1433 ∀wral 2348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 |
This theorem depends on definitions: df-bi 115 df-ral 2353 |
This theorem is referenced by: ral2imi 2427 trint 3890 peano2 4336 mpteqb 5282 lbzbi 8701 r19.29uz 9878 alzdvds 10254 |
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