Theorem r19.37 3067

Description: Restricted quantifier version of one direction of 19.37 2087. (The other direction does not 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 3065	. 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 2940	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
5	4	imim1i 61	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
6	1, 5	sylbi 206	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-ral 2901  df-rex 2902

This theorem is referenced by:  r19.37v  3068  ss2iundf  36970