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Theorem equsalhw 1742
Description: Weaker version of equsalh 1914 (requiring distinct variables) without using ax-12 1878. (Contributed by NM, 29-Nov-2015.)
Hypotheses
Ref Expression
equsalhw.1  |-  ( ps 
->  A. x ps )
equsalhw.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsalhw  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem equsalhw
StepHypRef Expression
1 equsalhw.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 sp 1728 . . . . . 6  |-  ( A. x ps  ->  ps )
3 equsalhw.1 . . . . . 6  |-  ( ps 
->  A. x ps )
42, 3impbii 180 . . . . 5  |-  ( A. x ps  <->  ps )
51, 4syl6bbr 254 . . . 4  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ps )
)
65pm5.74i 236 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  A. x ps ) )
76albii 1556 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  A. x ps )
)
83a1d 22 . . . 4  |-  ( ps 
->  ( x  =  y  ->  A. x ps )
)
93, 8alrimih 1555 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  A. x ps ) )
10 ax9v 1645 . . . . 5  |-  -.  A. x  -.  x  =  y
11 con3 126 . . . . . 6  |-  ( ( x  =  y  ->  A. x ps )  -> 
( -.  A. x ps  ->  -.  x  =  y ) )
1211al2imi 1551 . . . . 5  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ( A. x  -.  A. x ps 
->  A. x  -.  x  =  y ) )
1310, 12mtoi 169 . . . 4  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  -.  A. x  -.  A. x ps )
14 ax6o 1735 . . . 4  |-  ( -. 
A. x  -.  A. x ps  ->  ps )
1513, 14syl 15 . . 3  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
169, 15impbii 180 . 2  |-  ( ps  <->  A. x ( x  =  y  ->  A. x ps ) )
177, 16bitr4i 243 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  dvelimhw  1747  dvelimhwNEW7  29432  equsexv-x12  29735  equveliv  29737
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-11 1727
This theorem depends on definitions:  df-bi 177
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