Theorem stdpc4 1762

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 6	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
2	1	alimi 1442	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3		sb2 1754	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
4	2, 3	syl 14	1 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1340  [wsb 1749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-i9 1517  ax-ial 1521

This theorem depends on definitions:  df-bi 116  df-sb 1750

This theorem is referenced by:  sbh  1763  sbft  1835  pm13.183  2859  spsbc  2957