Theorem stdpc4 2100

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 3759 and rspsbc 3834. (Contributed by NM, 14-May-1993.) Revise df-sb 2093. (Revised by BJ, 22-Dec-2020.) Revise df-sb again. (Revised by Wolf Lammen, 4-Jun-2026.)

Assertion

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

Proof of Theorem stdpc4

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stdpc4lem 2099	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		stdpc4lem 2099	. 2 ⊢ (∀𝑥𝜑 → ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
3		df-sb 2093	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ∧ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))))
4	1, 2, 3	sylanbrc 592	1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1560  [wsb 2092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932

This theorem depends on definitions:  df-bi 209  df-an 400  df-sb 2093

This theorem is referenced by:  sbtALT  2102  2stdpc4  2103  spsbim  2107  sbv  2123  sbft  2306  sb2  2512  sbtrt  2548  vexwt  2747  pm13.183  3627  spsbc  3759  nd1  10547  nd2  10548  bj-sbft  37258  bj-ab0  37398  wl-cbvalsbi  38054  wl-nfsbtv  38085  sbtd  42833  axfrege58b  44481  pm10.14  44940  pm11.57  44970