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Theorem dvelimfALT2OLD 29125
Description: Proof of dvelimh 1904 using dveeq2 1880 (shown as the last hypothesis) instead of ax12o 1875. As a consequence, theorem a12study2 29134 shows that ax12o 1875 could be replaced by dveeq2 1880 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) Obsolete as of 1-Aug-2017 - NM.
Hypotheses
Ref Expression
dvelimfALT2OLD.1  |-  ( ph  ->  A. x ph )
dvelimfALT2OLD.2  |-  ( ps 
->  A. z ps )
dvelimfALT2OLD.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
dvelimfALT2OLD.4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Assertion
Ref Expression
dvelimfALT2OLD  |-  ( -. 
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 dvelimfALT2OLD
StepHypRef Expression
1 ax-17 1603 . . 3  |-  ( -. 
A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
2 hbn1 1704 . . . 4  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
3 dvelimfALT2OLD.4 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
4 dvelimfALT2OLD.1 . . . . 5  |-  ( ph  ->  A. x ph )
54a1i 10 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( ph  ->  A. x ph )
)
62, 3, 5hbimd 1721 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
71, 6hbald 1714 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
8 dvelimfALT2OLD.2 . . 3  |-  ( ps 
->  A. z ps )
9 dvelimfALT2OLD.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
108, 9equsalh 1901 . 2  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
1110albii 1553 . 2  |-  ( A. x A. z ( z  =  y  ->  ph )  <->  A. x ps )
127, 10, 113imtr3g 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 1527    = wceq 1623
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-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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