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Mirrors > Home > MPE Home > Th. List > sb8e | Structured version Visualization version GIF version |
Description: Substitution of variable in existential quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2365. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
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 2520 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 2243 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvex 2408 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∃wex 1771 Ⅎwnf 1775 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: 2sb8e 2569 sb8mo 2680 bnj985 32124 exlimddvfi 35281 pm11.58 40599 |
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