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Theorem ax12olem4 1975
Description: Lemma for ax12o 1976. (Contributed by Wolf Lammen, 8-Feb-2018.)
Hypotheses
Ref Expression
ax12olem4.1  |-  ( ph  ->  F/ x  y  =  w )
ax12olem4.2  |-  ( ps 
->  F/ x  z  =  w )
Assertion
Ref Expression
ax12olem4  |-  ( ph  ->  ( ps  ->  (
y  =  z  ->  A. x  y  =  z ) ) )
Distinct variable groups:    x, w    y, w    z, w    ph, w    ps, w
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem ax12olem4
StepHypRef Expression
1 ax12olem1 1972 . . . 4  |-  ( y  =  z  <->  A. w
( y  =  w  ->  z  =  w ) )
2 nfv 1626 . . . . 5  |-  F/ w
( ph  /\  ps )
3 ax12olem4.1 . . . . . . 7  |-  ( ph  ->  F/ x  y  =  w )
43adantr 452 . . . . . 6  |-  ( (
ph  /\  ps )  ->  F/ x  y  =  w )
5 ax12olem4.2 . . . . . . 7  |-  ( ps 
->  F/ x  z  =  w )
65adantl 453 . . . . . 6  |-  ( (
ph  /\  ps )  ->  F/ x  z  =  w )
74, 6nfimd 1823 . . . . 5  |-  ( (
ph  /\  ps )  ->  F/ x ( y  =  w  ->  z  =  w ) )
82, 7nfald 1867 . . . 4  |-  ( (
ph  /\  ps )  ->  F/ x A. w
( y  =  w  ->  z  =  w ) )
91, 8nfxfrd 1577 . . 3  |-  ( (
ph  /\  ps )  ->  F/ x  y  =  z )
109nfrd 1775 . 2  |-  ( (
ph  /\  ps )  ->  ( y  =  z  ->  A. x  y  =  z ) )
1110ex 424 1  |-  ( ph  ->  ( ps  ->  (
y  =  z  ->  A. x  y  =  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   F/wnf 1550
This theorem is referenced by:  ax12o  1976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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