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Theorem strcollnft 10109
 Description: Closed form of strcollnf 10110. Version of ax-strcoll 10107 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
Assertion
Ref Expression
strcollnft
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)

Proof of Theorem strcollnft
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 strcoll2 10108 . 2
2 nfnf1 1436 . . . . 5
32nfal 1468 . . . 4
43nfal 1468 . . 3
5 nfa2 1471 . . . 4
6 nfvd 1422 . . . . 5
7 nfa1 1434 . . . . . . . 8
8 nfcvd 2179 . . . . . . . 8
9 sp 1401 . . . . . . . 8
107, 8, 9nfrexdxy 2357 . . . . . . 7
1110sps 1430 . . . . . 6
1211alcoms 1365 . . . . 5
136, 12nfbid 1480 . . . 4
145, 13nfald 1643 . . 3
15 nfv 1421 . . . . . 6
165, 15nfan 1457 . . . . 5
17 elequ2 1601 . . . . . . 7
1817adantl 262 . . . . . 6
1918bibi1d 222 . . . . 5
2016, 19albid 1506 . . . 4
2120ex 108 . . 3
224, 14, 21cbvexd 1802 . 2
231, 22syl5ib 143 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98  wal 1241  wnf 1349  wex 1381  wral 2306  wrex 2307 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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-strcoll 10107 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312 This theorem is referenced by:  strcollnf  10110
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