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Theorem sscoll2 9448
Description: Version of ax-sscoll 9447 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
Assertion
Ref Expression
sscoll2  c  a  b  d  c  d  a
Distinct variable groups:    a, b, c, d,,,   , c, d
Allowed substitution hints:   (,,, a, b)

Proof of Theorem sscoll2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . 3  F/ c  a  b
2 nfv 1418 . . . 4  F/  a  b
3 simpl 102 . . . . . 6  a  b  a
4 rexeq 2500 . . . . . . 7  b  b
54adantl 262 . . . . . 6  a  b  b
63, 5raleqbidv 2511 . . . . 5  a  b  a  b
7 nfv 1418 . . . . . 6  F/ d  a  b
8 nfv 1418 . . . . . . 7  F/  a  b
9 rexeq 2500 . . . . . . . . 9  a  a
109adantr 261 . . . . . . . 8  a  b  a
1110bibi2d 221 . . . . . . 7  a  b  d  d  a
128, 11albid 1503 . . . . . 6  a  b  d  d  a
137, 12rexbid 2319 . . . . 5  a  b  d  c  d  d  c  d  a
146, 13imbi12d 223 . . . 4  a  b  d  c  d  a  b  d  c  d  a
152, 14albid 1503 . . 3  a  b  d  c  d  a  b  d  c  d  a
161, 15exbid 1504 . 2  a  b  c  d  c  d  c  a  b  d  c  d  a
17 ax-sscoll 9447 . . . 4  c  d  c  d
1817spi 1426 . . 3  c  d  c  d
1918spi 1426 . 2  c  d  c  d
2016, 19ch2varv 9243 1  c  a  b  d  c  d  a
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240  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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sscoll 9447
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: (None)
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