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Mirrors > Home > ILE Home > Th. List > reubiia | GIF version |
Description: Formula-building rule for restricted existential quantifier (inference form). (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 449 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) |
3 | 2 | eubii 2008 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
4 | df-reu 2423 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-reu 2423 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | 3, 4, 5 | 3bitr4i 211 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∃!weu 1999 ∃!wreu 2418 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-eu 2002 df-reu 2423 |
This theorem is referenced by: reubii 2616 |
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