Theorem stdpc4 2341

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 3415 and rspsbc 3484. (Contributed by NM, 14-May-1993.)

Assertion

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

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ala1 1755	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb2 2340	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	1, 2	syl 17	1 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1473  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868

This theorem is referenced by:  2stdpc4  2342  sbft  2367  spsbim  2382  spsbbi  2390  sbt  2407  sbtrt  2408  pm13.183  3313  spsbc  3415  nd1  9288  nd2  9289  bj-vexwt  32048  axfrege58b  37214  pm10.14  37580  pm11.57  37611