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Theorem stdpc7 1859
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1651.) 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 1633 . 2  |-  ( y  =  x  ->  ( [ x  /  y ] ph  ->  ph ) )
21equcoms 1652 1  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6   [wsb 1631
This theorem is referenced by:  ax16  1990  sbequi  1998  sb5rf  2029  sb8  2031
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645
This theorem depends on definitions:  df-bi 179  df-an 362  df-sb 1632
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