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Theorem equsexv-x12 29735
Description: Weaker version of equsex 1915 without using sp 1728, ax9 1902, ax10 1897, or ax12o but allowing ax9v 1645. Experiment to study ax12o 1887. (Contributed by NM, 7-Nov-1015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
equsexv-x12.1  |-  ( ps 
->  A. x ps )
equsexv-x12.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsexv-x12  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem equsexv-x12
StepHypRef Expression
1 exnal 1564 . 2  |-  ( E. x  -.  ( x  =  y  ->  -.  ph )  <->  -.  A. x
( x  =  y  ->  -.  ph ) )
2 df-an 360 . . 3  |-  ( ( x  =  y  /\  ph )  <->  -.  ( x  =  y  ->  -.  ph ) )
32exbii 1572 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x  -.  ( x  =  y  ->  -.  ph ) )
4 equsexv-x12.1 . . . . 5  |-  ( ps 
->  A. x ps )
54hbn 1732 . . . 4  |-  ( -. 
ps  ->  A. x  -.  ps )
6 equsexv-x12.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76notbid 285 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
85, 7equsalhw 1742 . . 3  |-  ( A. x ( x  =  y  ->  -.  ph )  <->  -. 
ps )
98con2bii 322 . 2  |-  ( ps  <->  -. 
A. x ( x  =  y  ->  -.  ph ) )
101, 3, 93bitr4i 268 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  equvinv  29736
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  df-an 360  df-ex 1532
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