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Theorem ax11v 1724
Description: This is a version of ax-11o 1720 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
Assertion
Ref Expression
ax11v  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem ax11v
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 a9e 1602 . 2  |-  E. z 
z  =  y
2 ax-17 1435 . . . . 5  |-  ( ph  ->  A. z ph )
3 ax-11 1413 . . . . 5  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
42, 3syl5 32 . . . 4  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
5 equequ2 1615 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
65imbi1d 224 . . . . . . 7  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
76albidv 1721 . . . . . 6  |-  ( z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x
( x  =  y  ->  ph ) ) )
87imbi2d 223 . . . . 5  |-  ( z  =  y  ->  (
( ph  ->  A. x
( x  =  z  ->  ph ) )  <->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
95, 8imbi12d 227 . . . 4  |-  ( z  =  y  ->  (
( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph ) ) )  <->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
104, 9mpbii 140 . . 3  |-  ( z  =  y  ->  (
x  =  y  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
1110exlimiv 1505 . 2  |-  ( E. z  z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
121, 11ax-mp 7 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-17 1435  ax-i9 1439
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  equs5or  1727  sb56  1781
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