Theorem stdpc4 2428

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 3665 and rspsbc 3735. For a version requiring disjoint variables, but fewer axioms, see stdpc4v 2244. (Contributed by NM, 14-May-1993.)

Assertion

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

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ala1 1857	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb2 2427	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	1, 2	syl 17	1 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1599  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012

This theorem is referenced by:  2stdpc4  2429  sbft  2455  spsbim  2470  spsbbi  2478  sbt  2496  sbtrt  2497  pm13.183OLD  3550  spsbc  3665  nd1  9746  nd2  9747  bj-vexwt  33427  axfrege58b  39160  pm10.14  39524  pm11.57  39555