Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sb8eh | GIF version |
Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) |
Ref | Expression |
---|---|
sb8eh.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
sb8eh | ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8eh.1 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | hbsb3 1729 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
3 | sbequ12 1694 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvexh 1678 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 ∃wex 1421 [wsb 1685 |
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 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-sb 1686 |
This theorem is referenced by: exsb 1925 |
Copyright terms: Public domain | W3C validator |