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Theorem strcollnft 14358
Description: Closed form of strcollnf 14359. (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 14357 . 2 (∀𝑥𝑎𝑦𝜑 → ∃𝑧(∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑))
2 nfnf1 1544 . . . . 5 𝑏𝑏𝜑
32nfal 1576 . . . 4 𝑏𝑦𝑏𝜑
43nfal 1576 . . 3 𝑏𝑥𝑦𝑏𝜑
5 nfa1 1541 . . . . 5 𝑥𝑥𝑦𝑏𝜑
6 nfcvd 2320 . . . . 5 (∀𝑥𝑦𝑏𝜑𝑏𝑎)
7 nfa1 1541 . . . . . . 7 𝑦𝑦𝑏𝜑
87nfal 1576 . . . . . 6 𝑦𝑥𝑦𝑏𝜑
9 nfcvd 2320 . . . . . 6 (∀𝑥𝑦𝑏𝜑𝑏𝑧)
10 sp 1511 . . . . . . 7 (∀𝑦𝑏𝜑 → Ⅎ𝑏𝜑)
1110sps 1537 . . . . . 6 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝜑)
128, 9, 11nfrexdxy 2511 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦𝑧 𝜑)
135, 6, 12nfraldxy 2510 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎𝑦𝑧 𝜑)
145, 6, 11nfrexdxy 2511 . . . . 5 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑥𝑎 𝜑)
158, 9, 14nfraldxy 2510 . . . 4 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏𝑦𝑧𝑥𝑎 𝜑)
1613, 15nfand 1568 . . 3 (∀𝑥𝑦𝑏𝜑 → Ⅎ𝑏(∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑))
17 nfv 1528 . . . . . . 7 𝑥 𝑧 = 𝑏
185, 17nfan 1565 . . . . . 6 𝑥(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
19 rexeq 2673 . . . . . . 7 (𝑧 = 𝑏 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑏 𝜑))
2019adantl 277 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑏 𝜑))
2118, 20ralbid 2475 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑥𝑎𝑦𝑧 𝜑 ↔ ∀𝑥𝑎𝑦𝑏 𝜑))
22 nfv 1528 . . . . . . 7 𝑦 𝑧 = 𝑏
238, 22nfan 1565 . . . . . 6 𝑦(∀𝑥𝑦𝑏𝜑𝑧 = 𝑏)
24 eleq2 2241 . . . . . . . 8 (𝑧 = 𝑏 → (𝑦𝑧𝑦𝑏))
2524adantl 277 . . . . . . 7 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (𝑦𝑧𝑦𝑏))
2625imbi1d 231 . . . . . 6 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((𝑦𝑧 → ∃𝑥𝑎 𝜑) ↔ (𝑦𝑏 → ∃𝑥𝑎 𝜑)))
2723, 26ralbid2 2481 . . . . 5 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → (∀𝑦𝑧𝑥𝑎 𝜑 ↔ ∀𝑦𝑏𝑥𝑎 𝜑))
2821, 27anbi12d 473 . . . 4 ((∀𝑥𝑦𝑏𝜑𝑧 = 𝑏) → ((∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
2928ex 115 . . 3 (∀𝑥𝑦𝑏𝜑 → (𝑧 = 𝑏 → ((∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))))
304, 16, 29cbvexd 1927 . 2 (∀𝑥𝑦𝑏𝜑 → (∃𝑧(∀𝑥𝑎𝑦𝑧 𝜑 ∧ ∀𝑦𝑧𝑥𝑎 𝜑) ↔ ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
311, 30imbitrid 154 1 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wnf 1460  wex 1492  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-strcoll 14356
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461
This theorem is referenced by:  strcollnf  14359
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