Theorem r19.35 3065

Description: Restricted quantifier version of 19.35 1794. (Contributed by NM, 20-Sep-2003.)

Assertion

Ref	Expression
r19.35	⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.35

Step	Hyp	Ref	Expression
1		r19.26 3046	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
2		annim 440	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
3	2	ralbii 2963	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓))
4		df-an 385	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜓) ↔ ¬ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
5	1, 3, 4	3bitr3i 289	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓) ↔ ¬ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
6	5	con2bii 346	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓))
7		dfrex2 2979	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓)
8	7	imbi2i 325	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
9		dfrex2 2979	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 → 𝜓))
10	6, 8, 9	3bitr4ri 292	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀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

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

This theorem is referenced by:  r19.36v  3066  r19.37  3067  r19.43  3074  r19.37zv  4019  r19.36zv  4024  iinexg  4751  bndndx  11168  nmobndseqi  27018  nmobndseqiALT  27019