Theorem stdpc7 2076

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2075.) 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]𝜑 → 𝜑))

Proof of Theorem stdpc7

Step	Hyp	Ref	Expression
1		sbequ2 2013	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 → 𝜑))
2	1	equcoms 2067	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑))

Syntax hints:   → wi 4  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012

This theorem is referenced by: (None)