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Mirrors > Home > ILE Home > Th. List > reubii | GIF version |
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.) |
Ref | Expression |
---|---|
reubii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
reubii | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reubii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | a1i 9 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) |
3 | 2 | reubiia 2494 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∈ wcel 1393 ∃!wreu 2308 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-eu 1903 df-reu 2313 |
This theorem is referenced by: caucvgsrlemcl 6873 axcaucvglemcl 6969 axcaucvglemval 6971 |
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