![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > stdpc4 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
stdpc4 | ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
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) |
Colors of variables: wff setvar class |
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 |
Copyright terms: Public domain | W3C validator |