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Mirrors > Home > ILE Home > Th. List > sb8e | GIF version |
Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Ref | Expression |
---|---|
sb8e.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8e | ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8e.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfs1 1734 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 1698 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvex 1683 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 Ⅎwnf 1392 ∃wex 1424 [wsb 1689 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-11 1440 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 |
This theorem depends on definitions: df-bi 115 df-nf 1393 df-sb 1690 |
This theorem is referenced by: sb8mo 1959 mo2n 1973 mor 1987 nfrexdya 2409 |
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