Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sb8e | Structured version Visualization version GIF version |
Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2372. For a version requiring disjoint variables, but fewer axioms, see sb8ef 2353. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb8.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8e | ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfs1 2492 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 2244 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvex 2399 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1782 Ⅎwnf 1786 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 df-sb 2068 |
This theorem is referenced by: 2sb8e 2535 sb8mo 2601 bnj985 32934 exlimddvfi 36280 pm11.58 42008 |
Copyright terms: Public domain | W3C validator |