Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  strcollnft GIF version

Theorem strcollnft 13518
 Description: Closed form of strcollnf 13519. (Contributed by BJ, 21-Oct-2019.)
Assertion
Ref Expression
strcollnft (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
Distinct variable group:   𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem strcollnft
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 strcoll2 13517 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧(∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑))
2 nfnf1 1524 . . . . 5 𝑏𝑏𝜑
32nfal 1556 . . . 4 𝑏𝑦𝑏𝜑
43nfal 1556 . . 3 𝑏𝑥𝑦𝑏𝜑
5 nfa1 1521 . . . . 5 𝑥𝑥𝑦𝑏𝜑
6 nfcvd 2300 . . . . 5 (∀𝑥𝑦𝑏𝜑𝑏𝑎)
7 nfa1 1521 . . . . . . 7 𝑦𝑦𝑏𝜑
87nfal 1556 . . . . . 6 𝑦𝑥𝑦𝑏𝜑
9 nfcvd 2300 . . . . . 6 (∀𝑥𝑦𝑏𝜑𝑏𝑧)
10 sp 1491 . . . . . . 7 (∀𝑦𝑏𝜑 → Ⅎ𝑏𝜑)
1110sps 1517 . . . . . 6 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝜑)
128, 9, 11nfrexdxy 2491 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦𝑧 𝜑)
135, 6, 12nfraldxy 2490 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎𝑦𝑧 𝜑)
145, 6, 11nfrexdxy 2491 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
158, 9, 14nfraldxy 2490 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦𝑧𝑥𝑎 𝜑)
1613, 15nfand 1548 . . 3 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏(∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑))
17 nfv 1508 . . . . . . 7 𝑥 𝑧 = 𝑏
185, 17nfan 1545 . . . . . 6 𝑥(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
19 rexeq 2653 . . . . . . 7 (𝑧 = 𝑏 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑏 𝜑))
2019adantl 275 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑏 𝜑))
2118, 20ralbid 2455 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑥𝑎𝑦𝑧 𝜑 ↔ ∀𝑥𝑎𝑦𝑏 𝜑))
22 nfv 1508 . . . . . . 7 𝑦 𝑧 = 𝑏
238, 22nfan 1545 . . . . . 6 𝑦(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
24 eleq2 2221 . . . . . . . 8 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
2524adantl 275 . . . . . . 7 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (𝑦𝑧𝑦𝑏))
2625imbi1d 230 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((𝑦𝑧 → ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 → ∃𝑥𝑎 𝜑)))
2723, 26ralbid2 2461 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑦𝑧𝑥𝑎 𝜑 ↔ ∀𝑦𝑏𝑥𝑎 𝜑))
2821, 27anbi12d 465 . . . 4 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
2928ex 114 . . 3 (∀𝑥𝑦𝑏𝜑 → (𝑧 = 𝑏 → ((∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))))
304, 16, 29cbvexd 1907 . 2 (∀𝑥𝑦𝑏𝜑 → (∃𝑧(∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑) ↔ ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
311, 30syl5ib 153 1 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1333  Ⅎwnf 1440  ∃wex 1472  ∀wral 2435  ∃wrex 2436 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-strcoll 13516 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441 This theorem is referenced by:  strcollnf  13519
 Copyright terms: Public domain W3C validator