![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sb1 | GIF version |
Description: One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sb1 | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 1774 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | 1 | simprbi 275 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃wex 1503 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 |
This theorem depends on definitions: df-bi 117 df-sb 1774 |
This theorem is referenced by: sbh 1787 sbiedh 1798 sb4a 1812 sb4e 1816 sbcof2 1821 sb4 1843 sb4or 1844 spsbe 1853 sbidm 1862 sb5rf 1863 bj-sbimedh 15001 |
Copyright terms: Public domain | W3C validator |