Theorem stdpc4 2024

Description: The specialization axiom of standard predicate calculus. It states that if a statement φ holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: "∀xφ(x) → φ(y), provided that y is free for x in φ(x)." Axiom 4 of [Mendelson] p. 69. See also spsbc 3059 and rspsbc 3125. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
stdpc4	⊢ (∀xφ → [y / x]φ)

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (φ → (x = y → φ))
2	1	alimi 1559	. 2 ⊢ (∀xφ → ∀x(x = y → φ))
3		sb2 2023	. 2 ⊢ (∀x(x = y → φ) → [y / x]φ)
4	2, 3	syl 15	1 ⊢ (∀xφ → [y / x]φ)

Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbft  2025  spsbe  2075  spsbim  2076  spsbbi  2077  sb8  2092  sb9i  2094  pm13.183  2980  spsbc  3059