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Mirrors > Home > MPE Home > Th. List > ral2imi | Structured version Visualization version GIF version |
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3087. (Revised by Wolf Lammen, 1-Dec-2019.) |
Ref | Expression |
---|---|
ral2imi.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ral2imi | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3063 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | ral2imi.1 | . . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 2 | imim3i 64 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → 𝜒))) |
4 | 3 | al2imi 1818 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 3063 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
6 | df-ral 3063 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
7 | 4, 5, 6 | 3imtr4g 296 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∈ wcel 2107 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ral 3063 |
This theorem is referenced by: ralim 3087 rexim 3088 ralbi 3104 r19.26 3112 iiner 8783 ss2ixp 8904 undifixp 8928 boxriin 8934 acni2 10041 axcc4 10434 intgru 10809 ingru 10810 prdsdsval3 17431 mndind 18709 hauscmplem 22910 uspgr2wlkeq 28934 wlkp1lem8 28968 prdstotbnd 36710 mnuunid 43084 |
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