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Mirrors > Home > ILE Home > Th. List > r19.27m | GIF version |
Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Ref | Expression |
---|---|
r19.27m.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
r19.27m | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.27m.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | r19.3rm 3330 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
3 | 2 | anbi2d 451 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
4 | r19.26 2485 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 3, 4 | syl6rbbr 197 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 Ⅎwnf 1389 ∃wex 1421 ∈ wcel 1433 ∀wral 2348 |
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-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-cleq 2074 df-clel 2077 df-ral 2353 |
This theorem is referenced by: r19.27mv 3337 raaanlem 3346 |
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