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Mirrors > Home > MPE Home > Th. List > sb8ev | Structured version Visualization version GIF version |
Description: Substitution of variable in existential quantifier. Version of sb8e 2537 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) |
Ref | Expression |
---|---|
sb8v.nf | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8ev | ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8v.nf | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfs1v 2157 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | sbequ12 2250 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvexv1 2351 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1781 Ⅎwnf 1785 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 |
This theorem is referenced by: 2sb8ev 2364 sbnf2 2366 mo3 2623 cbvmowOLD 2664 bnj985v 32335 sbcexf 35553 |
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