Theorem stdpc7 1176

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1123.) 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.

Assertion

Ref	Expression
stdpc7	|- (x = y -> ([x / y]ph -> ph))

Proof of Theorem stdpc7

Step	Hyp	Ref	Expression
1		sbequ2 1175	. 2 |- (y = x -> ([x / y]ph -> ph))
2	1	equcoms 1126	1 |- (x = y -> ([x / y]ph -> ph))

Syntax hints:   -> wi 3   = wceq 953  [wsbc 1166

This theorem is referenced by:  ax16 1205  sbequi 1223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-8 961  ax-12 965  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119

This theorem depends on definitions:  df-bi 147  df-an 225  df-sb 1168

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