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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 2367. For a version requiring disjoint variables, but fewer axioms, see sb8ef 2347. (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 2483 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 2239 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvex 2394 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1774 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: 2sb8e 2525 sb8mo 2591 bnj985 34579 exlimddvfi 37589 pm11.58 43821 |
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