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Theorem aev 1690
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1692. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
aev
Distinct variable group:   ,

Proof of Theorem aev
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae 1603 . 2
2 hbae 1603 . . . 4
3 ax-8 1392 . . . . 5
43spimv 1689 . . . 4
52, 4alrimih 1355 . . 3
6 ax-8 1392 . . . . . . . 8
7 equcomi 1589 . . . . . . . 8
86, 7syl6 29 . . . . . . 7
98spimv 1689 . . . . . 6
109alequcoms 1406 . . . . 5
1110a5i 1432 . . . 4
12 hbae 1603 . . . . 5
13 ax-8 1392 . . . . . 6
1413spimv 1689 . . . . 5
1512, 14alrimih 1355 . . . 4
16 alequcom 1405 . . . 4
1711, 15, 163syl 17 . . 3
18 ax-8 1392 . . . 4
1918spimv 1689 . . 3
205, 17, 193syl 17 . 2
211, 20alrimih 1355 1
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240
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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  ax16  1691  a16g  1741
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