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Mirrors > Home > NFE Home > Th. List > ralidm | GIF version |
Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
Ref | Expression |
---|---|
ralidm | ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rzal 3651 | . . 3 ⊢ (A = ∅ → ∀x ∈ A ∀x ∈ A φ) | |
2 | rzal 3651 | . . 3 ⊢ (A = ∅ → ∀x ∈ A φ) | |
3 | 1, 2 | 2thd 231 | . 2 ⊢ (A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)) |
4 | neq0 3560 | . . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A) | |
5 | biimt 325 | . . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ))) | |
6 | df-ral 2619 | . . . . 5 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x(x ∈ A → ∀x ∈ A φ)) | |
7 | nfra1 2664 | . . . . . 6 ⊢ Ⅎx∀x ∈ A φ | |
8 | 7 | 19.23 1801 | . . . . 5 ⊢ (∀x(x ∈ A → ∀x ∈ A φ) ↔ (∃x x ∈ A → ∀x ∈ A φ)) |
9 | 6, 8 | bitri 240 | . . . 4 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ)) |
10 | 5, 9 | syl6rbbr 255 | . . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)) |
11 | 4, 10 | sylbi 187 | . 2 ⊢ (¬ A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)) |
12 | 3, 11 | pm2.61i 156 | 1 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∀wral 2614 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: (None) |
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