Theorem r19.37 2761

Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
r19.37.1	⊢ Ⅎxφ

Assertion

Ref	Expression
r19.37	⊢ (∃x ∈ A (φ → ψ) → (φ → ∃x ∈ A ψ))

Proof of Theorem r19.37

Step	Hyp	Ref	Expression
1		r19.35 2759	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))
2		r19.37.1	. . . 4 ⊢ Ⅎxφ
3		ax-1 6	. . . 4 ⊢ (φ → (x ∈ A → φ))
4	2, 3	ralrimi 2696	. . 3 ⊢ (φ → ∀x ∈ A φ)
5	4	imim1i 54	. 2 ⊢ ((∀x ∈ A φ → ∃x ∈ A ψ) → (φ → ∃x ∈ A ψ))
6	1, 5	sylbi 187	1 ⊢ (∃x ∈ A (φ → ψ) → (φ → ∃x ∈ A ψ))

Syntax hints:   → wi 4  Ⅎwnf 1544   ∈ 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-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:  r19.37av  2762