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 3415 and rspsbc 3484. (Contributed by NM, 14-May-1993.) |
Ref | Expression |
---|---|
stdpc4 | ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
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) |
Colors of variables: wff setvar class |
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 |
Copyright terms: Public domain | W3C validator |