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Mirrors > Home > NFE 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 618 | . . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ)) |
3 | 2 | eubii 2213 | . 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!x(x ∈ A ∧ ψ)) |
4 | df-reu 2622 | . 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ)) | |
5 | df-reu 2622 | . 2 ⊢ (∃!x ∈ A ψ ↔ ∃!x(x ∈ A ∧ ψ)) | |
6 | 3, 4, 5 | 3bitr4i 268 | 1 ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ∃!weu 2204 ∃!wreu 2617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-eu 2208 df-reu 2622 |
This theorem is referenced by: reubii 2798 |
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