Theorem stdpc7 1784

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1717.) 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

Step	Hyp	Ref	Expression
1		sbequ2 1783	. 2 |- ( y = x -> ( [ x / y ] ph -> ph ) )
2	1	equcoms 1722	1 |- ( x = y -> ( [ x / y ] ph -> ph ) )

Syntax hints:   -> wi 4   [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1463  ax-ie2 1508  ax-8 1518  ax-17 1540  ax-i9 1544

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  ax16  1827  sbequi  1853  sb5rf  1866