Theorem r19.23v 1282

Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers.

Assertion

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

Distinct variable group(s):   ψ,x

Proof of Theorem r19.23v

Step	Hyp	Ref	Expression
1		impexp 276	. . 3 ⊢ (((x ∈ A ∧ φ) → ψ) ↔ (x ∈ A → (φ → ψ)))
2	1	bial 695	. 2 ⊢ (∀x((x ∈ A ∧ φ) → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
3		df-rex 1206	. . . 4 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4	3	imbi1i 161	. . 3 ⊢ ((∃x ∈ A φ → ψ) ↔ (∃x(x ∈ A ∧ φ) → ψ))
5		19.23v 950	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → ψ) ↔ (∃x(x ∈ A ∧ φ) → ψ))
6	4, 5	bitr4 154	. 2 ⊢ ((∃x ∈ A φ → ψ) ↔ ∀x((x ∈ A ∧ φ) → ψ))
7		df-ral 1205	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
8	2, 6, 7	3bitr4r 159	1 ⊢ (∀x ∈ A (φ → ψ) ↔ (∃x ∈ A φ → ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  reluni 2493  ac6lem 3575  kmlem11 3590

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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