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Mirrors > Home > ILE Home > Th. List > r19.21v | GIF version |
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.21v | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.21 2511 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wral 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-4 1488 ax-17 1507 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-ral 2422 |
This theorem is referenced by: r19.32vdc 2583 rmo4 2881 rmo3 3004 dftr5 4037 reusv3 4389 tfrlem1 6213 tfrlemi1 6237 tfr1onlemaccex 6253 tfrcllemaccex 6266 tfri3 6272 ordiso2 6928 raluz2 9401 ndvdssub 11663 nninfalllem1 13378 nninfsellemqall 13386 |
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