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]𝜑 → 𝜑))

Proof of Theorem stdpc7

Step	Hyp	Ref	Expression
1		sbequ2 1652	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 → 𝜑))
2	1	equcoms 1594	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑))

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