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Theorem ax9lem16 29777
Description: Lemma for ax9 1902. Similar to ax10 1897 but with distinct variables, without using sp 1728, ax9 1902, or ax10 1897. We used ax9lem6 29767 to eliminate 5 hypotheses that would otherwise be needed. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem16.b  |-  -.  A. v  -.  v  =  x
ax9lem16.c  |-  -.  A. v  -.  v  =  y
ax9lem16.f  |-  -.  A. w  -.  w  =  x
ax9lem16.g  |-  -.  A. w  -.  w  =  z
ax9lem16.i  |-  -.  A. x  -.  x  =  w
ax9lem16.k  |-  -.  A. x  -.  x  =  z
ax9lem16.l  |-  -.  A. y  -.  y  =  v
ax9lem16.m  |-  -.  A. y  -.  y  =  w
ax9lem16.n  |-  -.  A. z  -.  z  =  v
ax9lem16.o  |-  -.  A. z  -.  z  =  w
Assertion
Ref Expression
ax9lem16  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Distinct variable group:    x, w, y, z, v

Proof of Theorem ax9lem16
StepHypRef Expression
1 ax9lem16.f . . 3  |-  -.  A. w  -.  w  =  x
2 ax9lem16.k . . 3  |-  -.  A. x  -.  x  =  z
3 ax9lem16.i . . 3  |-  -.  A. x  -.  x  =  w
41, 2, 3ax9lem15 29776 . 2  |-  ( A. x  x  =  y  ->  A. x  x  =  z )
5 ax9lem16.b . . . . 5  |-  -.  A. v  -.  v  =  x
6 ax9lem16.n . . . . . 6  |-  -.  A. z  -.  z  =  v
71, 3, 2, 6ax9lem6 29767 . . . . 5  |-  -.  A. x  -.  x  =  v
8 ax9lem16.c . . . . . 6  |-  -.  A. v  -.  v  =  y
91, 3, 7, 8ax9lem6 29767 . . . . 5  |-  -.  A. x  -.  x  =  y
10 ax9lem16.l . . . . 5  |-  -.  A. y  -.  y  =  v
11 ax9lem16.m . . . . 5  |-  -.  A. y  -.  y  =  w
125, 7, 9, 8, 10, 11ax9lem14 29775 . . . 4  |-  ( A. x  x  =  z  ->  A. w  w  =  z )
1310, 8, 5, 2ax9lem6 29767 . . . . . 6  |-  -.  A. v  -.  v  =  z
146, 13, 5, 3ax9lem6 29767 . . . . 5  |-  -.  A. v  -.  v  =  w
15 ax9lem16.o . . . . . 6  |-  -.  A. z  -.  z  =  w
16 ax9lem16.g . . . . . 6  |-  -.  A. w  -.  w  =  z
1715, 16, 1, 7ax9lem6 29767 . . . . 5  |-  -.  A. w  -.  w  =  v
1814, 1, 17ax9lem15 29776 . . . 4  |-  ( A. w  w  =  z  ->  A. w  w  =  x )
1912, 18syl 15 . . 3  |-  ( A. x  x  =  z  ->  A. w  w  =  x )
2015, 16, 17, 6, 13, 8ax9lem14 29775 . . 3  |-  ( A. w  w  =  x  ->  A. y  y  =  x )
2119, 20syl 15 . 2  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
224, 21syl 15 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax9lem17  29778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-ex 1532
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