Theorem r19.22 1272

Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.)

Assertion

Ref	Expression
r19.22	⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))

Proof of Theorem r19.22

Step	Hyp	Ref	Expression
1		imdistan 339	. . . 4 ⊢ ((x ∈ A → (φ → ψ)) ↔ ((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)))
2	1	bial 695	. . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ ∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)))
3		19.22 722	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)) → (∃x(x ∈ A ∧ φ) → ∃x(x ∈ A ∧ ψ)))
4	2, 3	sylbi 174	. 2 ⊢ (∀x(x ∈ A → (φ → ψ)) → (∃x(x ∈ A ∧ φ) → ∃x(x ∈ A ∧ ψ)))
5		df-ral 1205	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
6		df-rex 1206	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
7		df-rex 1206	. . 3 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
8	6, 7	imbi12i 163	. 2 ⊢ ((∃x ∈ A φ → ∃x ∈ A ψ) ↔ (∃x(x ∈ A ∧ φ) → ∃x(x ∈ A ∧ ψ)))
9	4, 5, 8	3imtr4 192	1 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))

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

This theorem is referenced by:  r19.22i 1273  r19.22d 1276  negeu 4124  receu 4215

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-gen 677

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

metamath.org