Theorem r19.21t 3260

Description: Restricted quantifier version of 19.21t 2245; closed form of r19.21 3261. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)

Assertion

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

Proof of Theorem r19.21t

Step	Hyp	Ref	Expression
1		19.21t 2245	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))))
2		df-ral 3081	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3		bi2.04 392	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
4	3	albii 1851	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
5	2, 4	bitri 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
6		df-ral 3081	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
7	6	imbi2i 339	. 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
8	1, 5, 7	3bitr4g 317	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  Ⅎwnf 1815   ∈ wcel 2146  ∀wral 3080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-12 2215

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816  df-ral 3081

This theorem is used by:  r19.21  3261