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

Theorem aev-o 2260
 Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2222. Version of aev 2048 using ax-10o 2217. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aev-o
Distinct variable group:   ,

Proof of Theorem aev-o
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbae-o 2231 . 2
2 hbae-o 2231 . . . 4
3 ax-8 1688 . . . . 5
43spimv 1964 . . . 4
52, 4alrimih 1575 . . 3
6 ax-8 1688 . . . . . . . 8
7 equcomi 1692 . . . . . . . 8
86, 7syl6 32 . . . . . . 7
98spimv 1964 . . . . . 6
109aecoms-o 2230 . . . . 5
1110a5i-o 2228 . . . 4
12 hbae-o 2231 . . . . 5
13 ax-8 1688 . . . . . 6
1413spimv 1964 . . . . 5
1512, 14alrimih 1575 . . . 4
16 aecom-o 2229 . . . 4
1711, 15, 163syl 19 . . 3
18 ax-8 1688 . . . 4
1918spimv 1964 . . 3
205, 17, 193syl 19 . 2
211, 20alrimih 1575 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1550 This theorem is referenced by:  a16g-o  2264 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-4 2213  ax-5o 2214  ax-6o 2215  ax-10o 2217  ax-12o 2220 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
 Copyright terms: Public domain W3C validator