Theorem r19.23t 3228

Description: Closed theorem form of r19.23 3229. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Assertion

Ref	Expression
r19.23t	⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)))

Proof of Theorem r19.23t

Step	Hyp	Ref	Expression
1		19.23t 2213	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)))
2		df-ral 3048	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3		impexp 450	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
4	3	albii 1820	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
5	2, 4	bitr4i 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
6		df-rex 3057	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	imbi1i 349	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
8	1, 5, 7	3bitr4g 314	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-ral 3048  df-rex 3057

This theorem is referenced by:  r19.23  3229  rexlimd2  3238  riotasv3d  38999