Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax16i Structured version   Unicode version

Theorem ax16i 2055
 Description: Inference with ax16 2054 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ax16i.1
ax16i.2
Assertion
Ref Expression
ax16i
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)   (,,)

Proof of Theorem ax16i
StepHypRef Expression
1 nfv 1631 . . 3
2 nfv 1631 . . 3
3 ax-8 1690 . . 3
41, 2, 3cbv3 1975 . 2
5 ax-8 1690 . . . . 5
65spimv 1967 . . . 4
7 equcomi 1694 . . . . . 6
8 equcomi 1694 . . . . . . 7
9 ax-8 1690 . . . . . . 7
108, 9syl 16 . . . . . 6
117, 10syl5com 29 . . . . 5
1211alimdv 1633 . . . 4
136, 12mpcom 35 . . 3
14 equcomi 1694 . . . 4
1514alimi 1569 . . 3
1613, 15syl 16 . 2
17 ax16i.1 . . . . 5
1817biimpcd 217 . . . 4
1918alimdv 1633 . . 3
20 ax16i.2 . . . . 5
2120nfi 1561 . . . 4
22 nfv 1631 . . . 4
2317biimprd 216 . . . . 5
2414, 23syl 16 . . . 4
2521, 22, 24cbv3 1975 . . 3
2619, 25syl6com 34 . 2
274, 16, 263syl 19 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178  wal 1550 This theorem is referenced by:  ax16ALT  2161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
 Copyright terms: Public domain W3C validator