Theorem stdpc4 1184

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 a4sbc 1942 and ra4sbc 1994.

Assertion

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

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ax-1 4	. . 3 ⊢ (φ → (x = y → φ))
2	1	19.20i 991	. 2 ⊢ (∀xφ → ∀x(x = y → φ))
3		sb2 1176	. 2 ⊢ (∀x(x = y → φ) → [y / x]φ)
4	2, 3	syl 10	1 ⊢ (∀xφ → [y / x]φ)

Syntax hints:   → wi 3  ∀wal 953  [wsbc 1169

This theorem is referenced by:  sbf 1185  hbs1f 1188  a4sbe 1242  a4sbim 1243  a4sbbi 1244  sb8 1260  sb9i 1262  a4sbc 1942  ra4sbc 1994  nd1 4921  nd2 4922

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171

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