Theorem stdpc4 869

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. Axiom 4 of [Mendelson] p. 59.

Assertion

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

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ax-1 3	. . 3 ⊢ (φ → (x = y → φ))
2	1	19.20i 691	. 2 ⊢ (∀xφ → ∀x(x = y → φ))
3		sb2 859	. 2 ⊢ (∀x(x = y → φ) → [y / x]φ)
4	2, 3	syl 12	1 ⊢ (∀xφ → [y / x]φ)

Syntax hints:   → wi 2  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  sbf 870  hbs1f 874  sbea4 894  sbia4 895  sbba4 896  sb8 918  sb9i 920  a4sbc 1444  nd1 3732  nd2 3733

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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