Theorem stdpc7 1958

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1957.) 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

Step	Hyp	Ref	Expression
1		sbequ2 1882	. 2 |- ( y = x -> ( [ x / y ] ph -> ph ) )
2	1	equcoms 1947	1 |- ( x = y -> ( [ x / y ] ph -> ph ) )

Syntax hints:   -> wi 4   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by: (None)