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Theorem stdpc7 1873
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1872.) Translated to traditional notation, it can be read: " x  =  y  ->  ( ph (
x ,  x )  ->  ph ( x ,  y ) ), provided that  y is free for  x in  ph ( x ,  x
)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
Assertion
Ref Expression
stdpc7  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )

Proof of Theorem stdpc7
StepHypRef Expression
1 sbequ2 1801 . 2  |-  ( y  =  x  ->  ( [ x  /  y ] ph  ->  ph ) )
21equcoms 1866 1  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   [wsb 1799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-sb 1800
This theorem is referenced by: (None)
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