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Theorem ax7 1871
Description: Proof of ax-7 1862 from ax7v1 1864 and ax7v2 1865, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1863, which is itself a weakened version of ax-7 1862.

Note that the weakened version of ax-7 1862 obtained by adding a dv condition on  x ,  z (resp. on  y ,  z) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

Assertion
Ref Expression
ax7  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )

Proof of Theorem ax7
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 ax7v2 1865 . . . 4  |-  ( x  =  t  ->  (
x  =  y  -> 
t  =  y ) )
2 ax7v2 1865 . . . 4  |-  ( x  =  t  ->  (
x  =  z  -> 
t  =  z ) )
3 ax7v1 1864 . . . . . 6  |-  ( t  =  y  ->  (
t  =  z  -> 
y  =  z ) )
43imp 435 . . . . 5  |-  ( ( t  =  y  /\  t  =  z )  ->  y  =  z )
54a1i 11 . . . 4  |-  ( x  =  t  ->  (
( t  =  y  /\  t  =  z )  ->  y  =  z ) )
61, 2, 5syl2and 490 . . 3  |-  ( x  =  t  ->  (
( x  =  y  /\  x  =  z )  ->  y  =  z ) )
76expd 442 . 2  |-  ( x  =  t  ->  (
x  =  y  -> 
( x  =  z  ->  y  =  z ) ) )
8 ax6evr 1870 . 2  |-  E. t  x  =  t
97, 8exlimiiv 1788 1  |-  ( x  =  y  ->  (
x  =  z  -> 
y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675
This theorem is referenced by:  equcomi  1872  equtr  1876  equequ1  1878  cbvaev  1897  aev  2037  aevALT  2166  axc16i  2167  equveli  2191  axext3  2444  dtru  4611  axextnd  9047  2spotmdisj  25852  bj-aev  31400  bj-dtru  31458  bj-mo3OLD  31493  wl-aetr  31909  wl-exeq  31913  wl-aleq  31914  wl-nfeqfb  31916  hbequid  32526  equidqe  32539  aev-o  32548  ax6e2eq  36969  ax6e2eqVD  37345
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