Theorem r19.23t 2601

Description: Closed theorem form of r19.23 2602. (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 1688	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)))
2		df-ral 2477	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3		impexp 263	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
4	3	albii 1481	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
5	2, 4	bitr4i 187	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
6		df-rex 2478	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	imbi1i 238	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
8	1, 5, 7	3bitr4g 223	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362  Ⅎwnf 1471  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  r19.23  2602  rexlimd2  2609