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Theorem stdpc7 1942
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1699.) 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 1660 . 2  |-  ( y  =  x  ->  ( [ x  /  y ] ph  ->  ph ) )
21equcoms 1693 1  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [wsb 1658
This theorem is referenced by:  sbequiOLD  2114  ax16ALT2  2155  sb5rf  2166  sbequiNEW7  29579  sb8iwAUX7  29589  sb8dwAUX7  29590  sb5rfNEW7  29591  sb8wAUX7  29593  ax16ALT2OLD7  29743  sb8OLD7  29756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659
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