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Mirrors > Home > ILE Home > Th. List > ralrimd | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.) |
Ref | Expression |
---|---|
ralrimd.1 | ⊢ Ⅎ𝑥𝜑 |
ralrimd.2 | ⊢ Ⅎ𝑥𝜓 |
ralrimd.3 | ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒))) |
Ref | Expression |
---|---|
ralrimd | ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralrimd.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | ralrimd.3 | . . 3 ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒))) | |
4 | 1, 2, 3 | alrimd 1501 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 2311 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
6 | 4, 5 | syl6ibr 151 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 Ⅎwnf 1349 ∈ wcel 1393 ∀wral 2306 |
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 1336 ax-gen 1338 ax-4 1400 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 |
This theorem is referenced by: ralrimdv 2398 fliftfun 5436 fzrevral 8967 |
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