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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 1508 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.21 2506 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wral 2414 |
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 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-ral 2419 |
This theorem is referenced by: r19.32vdc 2578 rmo4 2872 rmo3 2995 dftr5 4024 reusv3 4376 tfrlem1 6198 tfrlemi1 6222 tfr1onlemaccex 6238 tfrcllemaccex 6251 tfri3 6257 ordiso2 6913 raluz2 9367 ndvdssub 11616 nninfalllem1 13192 nninfsellemqall 13200 |
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