Theorem r19.36av 2760

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. (Contributed by NM, 22-Oct-2003.)

Assertion

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

Distinct variable group:   ψ,x

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem r19.36av

Step	Hyp	Ref	Expression
1		r19.35 2759	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))
2		idd 21	. . . 4 ⊢ (x ∈ A → (ψ → ψ))
3	2	rexlimiv 2733	. . 3 ⊢ (∃x ∈ A ψ → ψ)
4	3	imim2i 13	. 2 ⊢ ((∀x ∈ A φ → ∃x ∈ A ψ) → (∀x ∈ A φ → ψ))
5	1, 4	sylbi 187	1 ⊢ (∃x ∈ A (φ → ψ) → (∀x ∈ A φ → ψ))

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  iinss  4018