Theorem stdpc4 2075

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 𝑡 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "∀𝑥𝜑(𝑥) → 𝜑(𝑡), provided that 𝑡 is free for 𝑥 in 𝜑(𝑥)". Axiom 4 of [Mendelson] p. 69. See also spsbc 3733 and rspsbc 3817. (Contributed by NM, 14-May-1993.) Revise df-sb 2072. (Revised by BJ, 22-Dec-2020.)

Assertion

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

Proof of Theorem stdpc4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ala1 1820	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	a1d 25	. . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	alrimiv 1934	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		df-sb 2072	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	3, 4	sylibr 233	1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1540  [wsb 2071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 206  df-sb 2072

This theorem is referenced by:  sbtALT  2076  2stdpc4  2077  spsbim  2079  sbv  2095  sbft  2266  sb2  2482  sbtrt  2521  vexwt  2722  pm13.183  3599  spsbc  3733  nd1  10342  nd2  10343  bj-sbft  34951  bj-ab0  35087  wl-cbvalsbi  35698  sbtd  40171  axfrege58b  41476  pm10.14  41945  pm11.57  41975