Theorem r19.35 3343

Description: Restricted quantifier version of 19.35 1878. (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		pm2.27 42	. . . . 5 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	ralimi 3162	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓))
3		rexim 3243	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 → 𝜓) → 𝜓) → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓))
4	2, 3	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓))
5	4	com12 32	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
6		rexnal 3240	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
7		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
8	7	reximi 3245	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
9	6, 8	sylbir 237	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
10		ax-1 6	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
11	10	reximi 3245	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
12	9, 11	ja 188	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
13	5, 12	impbii 211	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wral 3140  ∃wrex 3141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-ral 3145  df-rex 3146

This theorem is referenced by:  r19.36v  3344  r19.37  3345  r19.37v  3346  r19.43  3353  r19.37zv  4449  r19.36zv  4454  iinexg  5246  bndndx  11899  nmobndseqi  28558  nmobndseqiALT  28559  r19.36vf  41411