Theorem stdpc7 1700

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1636.) 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 1699	. 2 |- ( y = x -> ( [ x / y ] ph -> ph ) )
2	1	equcoms 1641	1 |- ( x = y -> ( [ x / y ] ph -> ph ) )

Syntax hints:   -> wi 4   [wsb 1692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-gen 1383  ax-ie2 1428  ax-8 1440  ax-17 1464  ax-i9 1468

This theorem depends on definitions:  df-bi 115  df-sb 1693

This theorem is referenced by:  ax16  1741  sbequi  1767  sb5rf  1780