Theorem stdpc4 2071

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 3729 and rspsbc 3812. (Contributed by NM, 14-May-1993.) Revise df-sb 2068. (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 1816	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	a1d 25	. . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	alrimiv 1930	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		df-sb 2068	. 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 1537  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-sb 2068

This theorem is referenced by:  sbtALT  2072  2stdpc4  2073  spsbim  2075  sbv  2091  sbft  2262  sb2  2480  sbtrt  2519  vexwt  2720  pm13.183  3597  spsbc  3729  nd1  10343  nd2  10344  bj-sbft  34957  bj-ab0  35093  wl-cbvalsbi  35704  sbtd  40176  axfrege58b  41508  pm10.14  41977  pm11.57  42007