Theorem stdpc4 2106

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 3755 and rspsbc 3830. (Contributed by NM, 14-May-1993.) Revise df-sb 2101. (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 1845	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	a1d 26	. . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	alrimiv 1961	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		dfsb 2102	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	3, 4	sylibr 237	1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  [wsb 2100

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101

This theorem is used by:  sbtALT  2107  2stdpc4  2108  spsbim  2110  sbv  2126  sbft  2306  sb2  2511  sbtrt  2547  vexwt  2746  pm13.183  3623  spsbc  3755  nd1  10685  nd2  10686  bj-sbft  37810  bj-ab0  37950  wl-cbvalsbi  38621  wl-nfsbtv  38652  sbtd  43406  axfrege58b  45043  pm10.14  45486  pm11.57  45516