Theorem r19.23 2730

Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)

Hypothesis

Ref	Expression
r19.23.1	⊢ Ⅎxψ

Assertion

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

Proof of Theorem r19.23

Step	Hyp	Ref	Expression
1		r19.23.1	. 2 ⊢ Ⅎxψ
2		r19.23t 2729	. 2 ⊢ (Ⅎxψ → (∀x ∈ A (φ → ψ) ↔ (∃x ∈ A φ → ψ)))
3	1, 2	ax-mp 5	1 ⊢ (∀x ∈ A (φ → ψ) ↔ (∃x ∈ A φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  ∀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-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.23v  2731  rexlimi  2732  rexlimd  2736