Theorem r19.21v 2451

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.)

Assertion

Ref	Expression
r19.21v	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.21v

Step	Hyp	Ref	Expression
1		nfv 1467	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.21 2450	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wral 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-ral 2365

This theorem is referenced by:  r19.32vdc  2517  rmo4  2809  rmo3  2931  dftr5  3945  reusv3  4295  tfrlem1  6087  tfrlemi1  6111  tfr1onlemaccex  6127  tfrcllemaccex  6140  tfri3  6146  ordiso2  6782  raluz2  9128  ndvdssub  11269  nninfalllem1  12171  nninfsellemqall  12179