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Mirrors > Home > ILE Home > Th. List > reubida | GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
reubida.1 | ⊢ Ⅎ𝑥𝜑 |
reubida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reubida | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reubida.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | reubida.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.32da 439 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 1, 3 | eubid 1948 | . 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | df-reu 2355 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-reu 2355 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 221 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 Ⅎwnf 1389 ∈ wcel 1433 ∃!weu 1941 ∃!wreu 2350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-eu 1944 df-reu 2355 |
This theorem is referenced by: reubidva 2536 |
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