Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  alxfr GIF version

Theorem alxfr 4193
 Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
Hypothesis
Ref Expression
alxfr.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
alxfr ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem alxfr
StepHypRef Expression
1 alxfr.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
21spcgv 2640 . . . . . 6 (𝐴𝐵 → (∀𝑥𝜑𝜓))
32com12 27 . . . . 5 (∀𝑥𝜑 → (𝐴𝐵𝜓))
43alimdv 1759 . . . 4 (∀𝑥𝜑 → (∀𝑦 𝐴𝐵 → ∀𝑦𝜓))
54com12 27 . . 3 (∀𝑦 𝐴𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓))
65adantr 261 . 2 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓))
7 nfa1 1434 . . . . . 6 𝑦𝑦𝜓
8 nfv 1421 . . . . . 6 𝑦𝜑
9 sp 1401 . . . . . . 7 (∀𝑦𝜓𝜓)
109, 1syl5ibrcom 146 . . . . . 6 (∀𝑦𝜓 → (𝑥 = 𝐴𝜑))
117, 8, 10exlimd 1488 . . . . 5 (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴𝜑))
1211alimdv 1759 . . . 4 (∀𝑦𝜓 → (∀𝑥𝑦 𝑥 = 𝐴 → ∀𝑥𝜑))
1312com12 27 . . 3 (∀𝑥𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑))
1413adantl 262 . 2 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑))
156, 14impbid 120 1 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 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)
 Copyright terms: Public domain W3C validator