Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2768 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)) |
3 | 2 | com12 30 | . . . . 5 ⊢ (∀𝑥𝜑 → (𝐴 ∈ 𝐵 → 𝜓)) |
4 | 3 | alimdv 1851 | . . . 4 ⊢ (∀𝑥𝜑 → (∀𝑦 𝐴 ∈ 𝐵 → ∀𝑦𝜓)) |
5 | 4 | com12 30 | . . 3 ⊢ (∀𝑦 𝐴 ∈ 𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓)) |
6 | 5 | adantr 274 | . 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓)) |
7 | nfa1 1521 | . . . . . 6 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
8 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
9 | sp 1488 | . . . . . . 7 ⊢ (∀𝑦𝜓 → 𝜓) | |
10 | 9, 1 | syl5ibrcom 156 | . . . . . 6 ⊢ (∀𝑦𝜓 → (𝑥 = 𝐴 → 𝜑)) |
11 | 7, 8, 10 | exlimd 1576 | . . . . 5 ⊢ (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴 → 𝜑)) |
12 | 11 | alimdv 1851 | . . . 4 ⊢ (∀𝑦𝜓 → (∀𝑥∃𝑦 𝑥 = 𝐴 → ∀𝑥𝜑)) |
13 | 12 | com12 30 | . . 3 ⊢ (∀𝑥∃𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑)) |
14 | 13 | adantl 275 | . 2 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑)) |
15 | 6, 14 | impbid 128 | 1 ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 = wceq 1331 ∃wex 1468 ∈ wcel 1480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |