Theorem r19.35 3271

Description: Restricted quantifier version of 19.35 1880. (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		rexim 3172	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓) → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓))
2		pm2.27 42	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
3	2	ralimi 3087	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓))
4	1, 3	syl11 33	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5		rexnal 3169	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
6		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
7	6	reximi 3178	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
8	5, 7	sylbir 234	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
9		ax-1 6	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
10	9	reximi 3178	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
11	8, 10	ja 186	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
12	4, 11	impbii 208	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-ral 3069  df-rex 3070

This theorem is referenced by:  r19.36v  3272  r19.37  3273  r19.37v  3274  r19.43  3280  r19.37zv  4432  r19.36zv  4437  iinexg  5265  bndndx  12232  nmobndseqi  29141  nmobndseqiALT  29142  r19.36vf  42685