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Theorem equsalhwOLD 1861
Description: Obsolete proof of equsalhw 1860 as of 28-Dec-2017. (Contributed by NM, 29-Nov-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
equsalhwOLD.1  |-  ( ps 
->  A. x ps )
equsalhwOLD.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsalhwOLD  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem equsalhwOLD
StepHypRef Expression
1 equsalhwOLD.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 sp 1763 . . . . . 6  |-  ( A. x ps  ->  ps )
3 equsalhwOLD.1 . . . . . 6  |-  ( ps 
->  A. x ps )
42, 3impbii 181 . . . . 5  |-  ( A. x ps  <->  ps )
51, 4syl6bbr 255 . . . 4  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ps )
)
65pm5.74i 237 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  A. x ps ) )
76albii 1575 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  A. x ps )
)
83a1d 23 . . . 4  |-  ( ps 
->  ( x  =  y  ->  A. x ps )
)
93, 8alrimih 1574 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  A. x ps ) )
10 ax9v 1667 . . . . 5  |-  -.  A. x  -.  x  =  y
11 con3 128 . . . . . 6  |-  ( ( x  =  y  ->  A. x ps )  -> 
( -.  A. x ps  ->  -.  x  =  y ) )
1211al2imi 1570 . . . . 5  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ( A. x  -.  A. x ps 
->  A. x  -.  x  =  y ) )
1310, 12mtoi 171 . . . 4  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  -.  A. x  -.  A. x ps )
14 ax6o 1766 . . . 4  |-  ( -. 
A. x  -.  A. x ps  ->  ps )
1513, 14syl 16 . . 3  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
169, 15impbii 181 . 2  |-  ( ps  <->  A. x ( x  =  y  ->  A. x ps ) )
177, 16bitr4i 244 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551
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