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| 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 2377. For a version requiring disjoint variables, but fewer axioms, see sb8ef 2358. (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 2493 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
| 3 | sbequ12 2251 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
| 4 | 1, 2, 3 | cbvex 2404 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∃wex 1779 Ⅎwnf 1783 [wsb 2064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: 2sb8e 2535 sb8mo 2601 bnj985 34968 exlimddvfi 38129 pm11.58 44409 |
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