![]() |
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 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 3811 and rspsbc 3895. (Contributed by NM, 14-May-1993.) Revise df-sb 2065. (Revised by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
stdpc4 | ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ala1 1811 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | a1d 25 | . . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | 2 | alrimiv 1926 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | df-sb 2065 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
5 | 3, 4 | sylibr 234 | 1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 |
This theorem depends on definitions: df-bi 207 df-sb 2065 |
This theorem is referenced by: sbtALT 2069 2stdpc4 2070 spsbim 2072 sbv 2088 sbft 2265 sb2 2481 sbtrt 2517 vexwt 2716 pm13.183 3676 spsbc 3811 nd1 10652 nd2 10653 bj-sbft 36689 bj-ab0 36822 wl-cbvalsbi 37448 wl-nfsbtv 37479 sbtd 42152 axfrege58b 43802 pm10.14 44268 pm11.57 44298 |
Copyright terms: Public domain | W3C validator |