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Mirrors > Home > ILE Home > Th. List > reubiia | GIF version |
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.) |
Ref | Expression |
---|---|
reubiia.1 | ⊢ (x ∈ A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
reubiia | ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reubiia.1 | . . . 4 ⊢ (x ∈ A → (φ ↔ ψ)) | |
2 | 1 | pm5.32i 427 | . . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ)) |
3 | 2 | eubii 1906 | . 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!x(x ∈ A ∧ ψ)) |
4 | df-reu 2307 | . 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ)) | |
5 | df-reu 2307 | . 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ)) | |
6 | 3, 4, 5 | 3bitr4i 201 | 1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ 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-tru 1245 df-nf 1347 df-eu 1900 df-reu 2307 |
This theorem is referenced by: reubii 2489 |
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