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Mirrors > Home > ILE Home > Th. List > reubidv | GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.) |
Ref | Expression |
---|---|
reubidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reubidv | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reubidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | adantr 270 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
3 | 2 | reubidva 2536 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∈ wcel 1433 ∃!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: reueqd 2559 sbcreug 2894 srpospr 6959 creur 8036 creui 8037 divalg2 10326 |
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