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Theorem ax9lem14 29153
Description: Change bound variable without using sp 1716, ax9 1889, or ax10 1884. (Contributed by NM, 22-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem14.a  |-  -.  A. z  -.  z  =  x
ax9lem14.b  |-  -.  A. x  -.  x  =  z
ax9lem14.c  |-  -.  A. x  -.  x  =  v
ax9lem14.d  |-  -.  A. z  -.  z  =  v
ax9lem14.e  |-  -.  A. v  -.  v  =  z
ax9lem14.f  |-  -.  A. v  -.  v  =  y
Assertion
Ref Expression
ax9lem14  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Distinct variable groups:    x, z,
v, w    y, z,
v, w

Proof of Theorem ax9lem14
StepHypRef Expression
1 ax9lem14.a . . 3  |-  -.  A. z  -.  z  =  x
2 ax9lem14.b . . 3  |-  -.  A. x  -.  x  =  z
3 ax9lem14.c . . 3  |-  -.  A. x  -.  x  =  v
4 ax-17 1603 . . 3  |-  ( x  =  w  ->  A. v  x  =  w )
5 ax-17 1603 . . 3  |-  ( v  =  w  ->  A. x  v  =  w )
6 ax-8 1643 . . 3  |-  ( x  =  v  ->  (
x  =  w  -> 
v  =  w ) )
71, 2, 3, 4, 5, 6ax9lem13 29152 . 2  |-  ( A. x  x  =  w  ->  A. v  v  =  w )
8 ax9lem14.d . . 3  |-  -.  A. z  -.  z  =  v
9 ax9lem14.e . . 3  |-  -.  A. v  -.  v  =  z
10 ax9lem14.f . . 3  |-  -.  A. v  -.  v  =  y
11 ax-17 1603 . . 3  |-  ( v  =  w  ->  A. y 
v  =  w )
12 ax-17 1603 . . 3  |-  ( y  =  w  ->  A. v 
y  =  w )
13 ax-8 1643 . . 3  |-  ( v  =  y  ->  (
v  =  w  -> 
y  =  w ) )
148, 9, 10, 11, 12, 13ax9lem13 29152 . 2  |-  ( A. v  v  =  w  ->  A. y  y  =  w )
157, 14syl 15 1  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax9lem16  29155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715
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