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Mirrors > Home > ILE Home > Th. List > reubidva | GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) |
Ref | Expression |
---|---|
reubidva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reubidva | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | reubidva.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | reubida 2615 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 ∃!wreu 2419 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-eu 2003 df-reu 2424 |
This theorem is referenced by: reubidv 2617 f1ofveu 5770 srpospr 7615 icoshftf1o 9804 divalgb 11658 |
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