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Theorem stdpc7 1653
 Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1591.) 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 1652 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
21equcoms 1594 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-17 1419  ax-i9 1423 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by:  ax16  1694  sbequi  1720  sb5rf  1732
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