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Theorem ax9lem2 29763
Description: Lemma for ax9 1902. Similar to equequ2 1669, without using sp 1728, ax9 1902, or ax10 1897. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem2.a  |-  -.  A. w  -.  w  =  z
ax9lem2.b  |-  -.  A. w  -.  w  =  x
Assertion
Ref Expression
ax9lem2  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
Distinct variable groups:    x, w    z, w

Proof of Theorem ax9lem2
StepHypRef Expression
1 ax9lem2.a . . . 4  |-  -.  A. w  -.  w  =  z
21ax9lem1 29762 . . 3  |-  ( z  =  x  ->  x  =  z )
3 ax-8 1661 . . . 4  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
43com12 27 . . 3  |-  ( x  =  y  ->  (
x  =  z  -> 
z  =  y ) )
52, 4syl5 28 . 2  |-  ( x  =  y  ->  (
z  =  x  -> 
z  =  y ) )
6 ax9lem2.b . . . 4  |-  -.  A. w  -.  w  =  x
76ax9lem1 29762 . . 3  |-  ( x  =  y  ->  y  =  x )
81ax9lem1 29762 . . 3  |-  ( z  =  y  ->  y  =  z )
9 ax-8 1661 . . . 4  |-  ( y  =  z  ->  (
y  =  x  -> 
z  =  x ) )
109com12 27 . . 3  |-  ( y  =  x  ->  (
y  =  z  -> 
z  =  x ) )
117, 8, 10syl2im 34 . 2  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
125, 11impbid 183 1  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  ax9lem18  29779
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
This theorem depends on definitions:  df-bi 177
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