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Theorem dvelimALT 2085
Description: Version of dvelim 1969 that doesn't use ax-10 2092. (See dvelimh 1917 for a version that doesn't use ax-11 1727.) (Contributed by NM, 17-May-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dvelimALT.1  |-  ( ph  ->  A. x ph )
dvelimALT.2  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
dvelimALT  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Distinct variable groups:    ps, z    x, z    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y)

Proof of Theorem dvelimALT
StepHypRef Expression
1 ax-17 1606 . . 3  |-  ( -. 
A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
2 ax16ALT 2000 . . . . 5  |-  ( A. x  x  =  z  ->  ( ( z  =  y  ->  ph )  ->  A. x ( z  =  y  ->  ph ) ) )
32a1d 22 . . . 4  |-  ( A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( ( z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) ) )
4 hbn1 1716 . . . . . . 7  |-  ( -. 
A. x  x  =  z  ->  A. x  -.  A. x  x  =  z )
5 hbn1 1716 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
64, 5hban 1748 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  A. x
( -.  A. x  x  =  z  /\  -.  A. x  x  =  y ) )
7 ax12o 1887 . . . . . . 7  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
) )
87imp 418 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( z  =  y  ->  A. x  z  =  y )
)
9 dvelimALT.1 . . . . . . 7  |-  ( ph  ->  A. x ph )
109a1i 10 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( ph  ->  A. x ph )
)
116, 8, 10hbimd 1733 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
1211ex 423 . . . 4  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( ( z  =  y  ->  ph )  ->  A. x ( z  =  y  ->  ph )
) ) )
133, 12pm2.61i 156 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
141, 13hbald 1726 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
15 ax-17 1606 . . 3  |-  ( ps 
->  A. z ps )
16 dvelimALT.2 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
1715, 16equsalh 1914 . 2  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
1817albii 1556 . 2  |-  ( A. x A. z ( z  =  y  ->  ph )  <->  A. x ps )
1914, 17, 183imtr3g 260 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530
This theorem is referenced by:  dveeq2-o16  2137
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  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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