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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 → (φ(x, x) → φ(x, y)), provided that y is free for x in φ(x, x)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
Assertion
Ref Expression
stdpc7 (x = y → ([x / y]φφ))

Proof of Theorem stdpc7
StepHypRef Expression
1 sbequ2 1650 . 2 (y = x → ([x / y]φφ))
21equcoms 1681 1 (x = y → ([x / y]φφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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
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