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Theorem dvelimhw 1747
Description: Proof of dvelimh 1917 without using ax-12 1878 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)
Hypotheses
Ref Expression
dvelimhw.1  |-  ( ph  ->  A. x ph )
dvelimhw.2  |-  ( ps 
->  A. z ps )
dvelimhw.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
dvelimhw.4  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
Assertion
Ref Expression
dvelimhw  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem dvelimhw
StepHypRef Expression
1 ax-17 1606 . . 3  |-  ( -. 
A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
2 hbn1 1716 . . . 4  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
3 equcomi 1664 . . . . 5  |-  ( z  =  y  ->  y  =  z )
4 dvelimhw.4 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
)
5 equcomi 1664 . . . . . 6  |-  ( y  =  z  ->  z  =  y )
65alimi 1549 . . . . 5  |-  ( A. x  y  =  z  ->  A. x  z  =  y )
73, 4, 6syl56 30 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
8 dvelimhw.1 . . . . 5  |-  ( ph  ->  A. x ph )
98a1i 10 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( ph  ->  A. x ph )
)
102, 7, 9hbimd 1733 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
111, 10hbald 1726 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
12 dvelimhw.2 . . 3  |-  ( ps 
->  A. z ps )
13 dvelimhw.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
1412, 13equsalhw 1742 . 2  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
1514albii 1556 . 2  |-  ( A. x A. z ( z  =  y  ->  ph )  <->  A. x ps )
1611, 14, 153imtr3g 260 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  ax12olem6  1885
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-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177
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