Theorem r19.23t 2729

Description: Closed theorem form of r19.23 2730. (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 1800	. 2 ⊢ (Ⅎxψ → (∀x((x ∈ A ∧ φ) → ψ) ↔ (∃x(x ∈ A ∧ φ) → ψ)))
2		df-ral 2620	. . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
3		impexp 433	. . . 4 ⊢ (((x ∈ A ∧ φ) → ψ) ↔ (x ∈ A → (φ → ψ)))
4	3	albii 1566	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
5	2, 4	bitr4i 243	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x((x ∈ A ∧ φ) → ψ))
6		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
7	6	imbi1i 315	. 2 ⊢ ((∃x ∈ A φ → ψ) ↔ (∃x(x ∈ A ∧ φ) → ψ))
8	1, 5, 7	3bitr4g 279	1 ⊢ (Ⅎxψ → (∀x ∈ A (φ → ψ) ↔ (∃x ∈ A φ → ψ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is used by:  r19.23  2730  rexlimd2  2737