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Theorem stdpc7 1695
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1632.) 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,  y )." 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 1694 . 2  |-  ( y  =  x  ->  ( [ x  /  y ] ph  ->  ph ) )
21equcoms 1636 1  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-gen 1379  ax-ie2 1424  ax-8 1436  ax-17 1460  ax-i9 1464
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by:  ax16  1736  sbequi  1762  sb5rf  1775
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