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Mirrors > Home > MPE Home > Th. List > ralidmOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ralidm 4506 as of 2-Sep-2024. (Contributed by NM, 28-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ralidmOLD | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rzal 4503 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑) | |
2 | rzal 4503 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | |
3 | 1, 2 | 2thd 265 | . 2 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
4 | neq0 4340 | . . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
5 | df-ral 3056 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) | |
6 | nfra1 3275 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
7 | 6 | 19.23 2196 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) |
8 | 5, 7 | bitri 275 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) |
9 | biimt 360 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))) | |
10 | 8, 9 | bitr4id 290 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
11 | 4, 10 | sylbi 216 | . 2 ⊢ (¬ 𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
12 | 3, 11 | pm2.61i 182 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3055 ∅c0 4317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-ral 3056 df-dif 3946 df-nul 4318 |
This theorem is referenced by: (None) |
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