Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cbvexdva | GIF version |
Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
cbvaldva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvexdva | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfvd 1516 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | cbvaldva.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
4 | 3 | ex 114 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
5 | 1, 2, 4 | cbvexd 1914 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∃wex 1479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: cbvrexdva2 2697 acexmid 5835 tfrlemi1 6291 ltexpri 7545 recexpr 7570 |
Copyright terms: Public domain | W3C validator |