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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 1538 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.28m 3524 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∃wex 1502 ∈ wcel 2158 ∀wral 2465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-cleq 2180 df-clel 2183 df-ral 2470 |
This theorem is referenced by: iinrabm 3961 iindif2m 3966 iinin2m 3967 xpiindim 4776 fintm 5413 ixpiinm 6737 neipsm 13925 |
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