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Theorem dvelimfALT2 1790
Description: Proof of dvelimf 1991 using dveeq2 1788 (shown as the last hypothesis) instead of ax-12 1490. This shows that ax-12 1490 could be replaced by dveeq2 1788 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
Hypotheses
Ref Expression
dvelimfALT2.1  |-  ( ph  ->  A. x ph )
dvelimfALT2.2  |-  ( ps 
->  A. z ps )
dvelimfALT2.3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
dvelimfALT2.4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
Assertion
Ref Expression
dvelimfALT2  |-  ( -. 
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 dvelimfALT2
StepHypRef Expression
1 ax-17 1507 . . 3  |-  ( -. 
A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
2 hbn1 1631 . . . 4  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
3 dvelimfALT2.4 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
4 dvelimfALT2.1 . . . . 5  |-  ( ph  ->  A. x ph )
54a1i 9 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( ph  ->  A. x ph )
)
62, 3, 5hbimd 1553 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( (
z  =  y  ->  ph )  ->  A. x
( z  =  y  ->  ph ) ) )
71, 6hbald 1468 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. z ( z  =  y  ->  ph )  ->  A. x A. z ( z  =  y  ->  ph ) ) )
8 dvelimfALT2.2 . . 3  |-  ( ps 
->  A. z ps )
9 dvelimfALT2.3 . . 3  |-  ( z  =  y  ->  ( ph 
<->  ps ) )
108, 9equsalh 1705 . 2  |-  ( A. z ( z  =  y  ->  ph )  <->  ps )
1110albii 1447 . 2  |-  ( A. x A. z ( z  =  y  ->  ph )  <->  A. x ps )
127, 10, 113imtr3g 203 1  |-  ( -. 
A. x  x  =  y  ->  ( ps  ->  A. x ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104   A.wal 1330    = wceq 1332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338
This theorem is referenced by: (None)
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