Theorem r19.23 2613

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 2612	. 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 105  Ⅎwnf 1482  ∀wral 2483  ∃wrex 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-ral 2488  df-rex 2489

This theorem is referenced by:  r19.23v  2614  rexlimi  2615  rexlimd  2619