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Theorem strcollnft 9414
Description: Closed form of strcollnf 9415. Version of ax-strcoll 9412 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  F/ b  a  b  b  a
Distinct variable group:    a, b,,
Allowed substitution hints:   (,, a, b)

Proof of Theorem strcollnft
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 strcoll2 9413 . 2  a  a
2 nfnf1 1433 . . . . 5  F/ b F/ b
32nfal 1465 . . . 4  F/ b F/ b
43nfal 1465 . . 3  F/ b F/ b
5 nfa2 1468 . . . 4  F/ F/ b
6 nfvd 1419 . . . . 5  F/ b  F/ b
7 nfa1 1431 . . . . . . . 8  F/ F/ b
8 nfcvd 2176 . . . . . . . 8  F/ b  F/_ b a
9 sp 1398 . . . . . . . 8  F/ b  F/ b
107, 8, 9nfrexdxy 2351 . . . . . . 7  F/ b  F/ b  a
1110sps 1427 . . . . . 6  F/ b  F/ b  a
1211alcoms 1362 . . . . 5  F/ b  F/ b  a
136, 12nfbid 1477 . . . 4  F/ b  F/ b  a
145, 13nfald 1640 . . 3  F/ b  F/ b  a
15 nfv 1418 . . . . . 6  F/  b
165, 15nfan 1454 . . . . 5  F/ F/ b  b
17 elequ2 1598 . . . . . . 7  b  b
1817adantl 262 . . . . . 6  F/ b  b  b
1918bibi1d 222 . . . . 5  F/ b  b  a  b  a
2016, 19albid 1503 . . . 4  F/ b  b  a  b  a
2120ex 108 . . 3  F/ b  b  a  b  a
224, 14, 21cbvexd 1799 . 2  F/ b  a  b  b  a
231, 22syl5ib 143 1  F/ b  a  b  b  a
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   F/wnf 1346  wex 1378  wral 2300  wrex 2301
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-strcoll 9412
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306
This theorem is referenced by:  strcollnf  9415
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