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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 3130. (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 3111 | . 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 1817 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 3111 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
6 | df-ral 3111 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
7 | 4, 5, 6 | 3imtr4g 299 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
8 | 1, 7 | sylbi 220 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-ral 3111 |
This theorem is referenced by: ralim 3130 ralbi 3135 r19.26 3137 rexim 3204 iiner 8352 ss2ixp 8457 undifixp 8481 boxriin 8487 acni2 9457 axcc4 9850 intgru 10225 ingru 10226 prdsdsval3 16750 mndind 17984 hauscmplem 22011 uspgr2wlkeq 27435 wlkp1lem8 27470 prdstotbnd 35232 mnuunid 40985 |
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