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Theorem stdpc7 1631
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1569.) 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, y)." 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 1630 . 2 (y = x → ([x / y]φφ))
21equcoms 1572 1 (x = y → ([x / y]φφ))
Colors of variables: wff set class
Syntax hints:  wi 4  [wsb 1623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1314  ax-ie2 1360  ax-8 1372  ax-17 1396  ax-i9 1400
This theorem depends on definitions:  df-bi 110  df-sb 1624
This theorem is referenced by:  ax16  1672  sbequi  1698  sb5rf  1710
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