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Mirrors > Home > ILE Home > Th. List > reubida | GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
reubida.1 | ⊢ Ⅎxφ |
reubida.2 | ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ)) |
Ref | Expression |
---|---|
reubida | ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reubida.1 | . . 3 ⊢ Ⅎxφ | |
2 | reubida.2 | . . . 4 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ)) | |
3 | 2 | pm5.32da 425 | . . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ))) |
4 | 1, 3 | eubid 1904 | . 2 ⊢ (φ → (∃!x(x ∈ A ∧ ψ) ↔ ∃!x(x ∈ A ∧ χ))) |
5 | df-reu 2307 | . 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ)) | |
6 | df-reu 2307 | . 2 ⊢ (∃!x ∈ A χ ↔ ∃!x(x ∈ A ∧ χ)) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 Ⅎwnf 1346 ∈ wcel 1390 ∃!weu 1897 ∃!wreu 2302 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-eu 1900 df-reu 2307 |
This theorem is referenced by: reubidva 2486 |
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