Theorem r19.23t 2423

Description: Closed theorem form of r19.23 2424. (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 1567	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)))
2		df-ral 2311	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3		impexp 250	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
4	3	albii 1359	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
5	2, 4	bitr4i 176	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
6		df-rex 2312	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	imbi1i 227	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
8	1, 5, 7	3bitr4g 212	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.23  2424  rexlimd2  2431