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Mirrors > Home > ILE Home > Th. List > r19.28mv | GIF version |
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Ref | Expression |
---|---|
r19.28mv | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1462 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.28m 3332 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∃wex 1422 ∈ wcel 1434 ∀wral 2349 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-cleq 2075 df-clel 2078 df-ral 2354 |
This theorem is referenced by: iinrabm 3742 iindif2m 3747 iinin2m 3748 xpiindim 4495 fintm 5100 |
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