Theorem stdpc7 1917

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1687.) 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 1650	. 2 |- (y = x -> ([x / y]ph -> ph))
2	1	equcoms 1681	1 |- (x = y -> ([x / y]ph -> ph))

Syntax hints:   -> wi 4  [wsb 1648

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649

This theorem is referenced by:  ax16ALT2  2048  sbequi  2059  sb5rf  2090  sb8  2092