Theorem stdpc4 1655

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. (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 5	. . 3 ⊢ (φ → (x = y → φ))
2	1	alimi 1341	. 2 ⊢ (∀xφ → ∀x(x = y → φ))
3		sb2 1647	. 2 ⊢ (∀x(x = y → φ) → [y / x]φ)
4	2, 3	syl 14	1 ⊢ (∀xφ → [y / x]φ)

Syntax hints:   → wi 4  ∀wal 1240  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  sbh  1656  sbft  1725  pm13.183  2675  spsbc  2769