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Mirrors > Home > ILE Home > Th. List > reubii | GIF version |
Description: Formula-building rule for restricted existential quantifier (inference form). (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 2674 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2159 ∃!wreu 2469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-17 1536 ax-ial 1544 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-eu 2040 df-reu 2474 |
This theorem is referenced by: caucvgsrlemcl 7805 axcaucvglemcl 7911 axcaucvglemval 7913 |
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