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Theorem stdpc7 2002
 Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2001.) 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 1939 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
21equcoms 1993 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 1937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938 This theorem is referenced by: (None)
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