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Mirrors > Home > ILE Home > Th. List > alxfr | GIF version |
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.) |
Ref | Expression |
---|---|
alxfr.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
alxfr | ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alxfr.1 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | spcgv 2640 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)) |
3 | 2 | com12 27 | . . . . 5 ⊢ (∀𝑥𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
4 | 3 | alimdv 1759 | . . . 4 ⊢ (∀𝑥𝜑 → (∀𝑦 𝐴 ∈ 𝐵 → ∀𝑦𝜓)) |
5 | 4 | com12 27 | . . 3 ⊢ (∀𝑦 𝐴 ∈ 𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓)) |
6 | 5 | adantr 261 | . 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓)) |
7 | nfa1 1434 | . . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
8 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
9 | sp 1401 | . . . . . . 7 ⊢ (∀𝑦𝜓 → 𝜓) | |
10 | 9, 1 | syl5ibrcom 146 | . . . . . 6 ⊢ (∀𝑦𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 7, 8, 10 | exlimd 1488 | . . . . 5 ⊢ (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴 → 𝜑)) |
12 | 11 | alimdv 1759 | . . . 4 ⊢ (∀𝑦𝜓 → (∀𝑥∃𝑦 𝑥 = 𝐴 → ∀𝑥𝜑)) |
13 | 12 | com12 27 | . . 3 ⊢ (∀𝑥∃𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑)) |
14 | 13 | adantl 262 | . 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑)) |
15 | 6, 14 | impbid 120 | 1 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: (None) |
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