Theorem r19.36av 1299

Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when A is empty.

Assertion

Ref	Expression
r19.36av	⊢ (∃x ∈ A (φ → ψ) → (∀x ∈ A φ → ψ))

Distinct variable group(s):   ψ,x

Proof of Theorem r19.36av

Step	Hyp	Ref	Expression
1		r19.35 1298	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))
2		idd 11	. . . 4 ⊢ (x ∈ A → (ψ → ψ))
3	2	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ A ψ → ψ)
4	3	syl3 18	. 2 ⊢ ((∀x ∈ A φ → ∃x ∈ A ψ) → (∀x ∈ A φ → ψ))
5	1, 4	sylbi 174	1 ⊢ (∃x ∈ A (φ → ψ) → (∀x ∈ A φ → ψ))

Syntax hints:   → wi 2   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  iinss 2025

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-or 197  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

metamath.org