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Theorem stdpc7 1955
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1954.) 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 1879 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
21equcoms 1944 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878
This theorem is referenced by: (None)
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