Theorem stdpc4 2482

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 3581 and rspsbc 3651. (Contributed by NM, 14-May-1993.)

Assertion

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

Proof of Theorem stdpc4

Step	Hyp	Ref	Expression
1		ala1 1882	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb2 2481	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	1, 2	syl 17	1 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1622  [wsb 2038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-12 2188  ax-13 2383

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1846  df-sb 2039

This theorem is referenced by:  2stdpc4  2483  sbft  2508  spsbim  2523  spsbbi  2531  sbt  2548  sbtrt  2549  pm13.183  3476  spsbc  3581  nd1  9593  nd2  9594  bj-vexwt  33152  axfrege58b  38688  pm10.14  39052  pm11.57  39083