Theorem r19.35 3121

Description: Restricted quantifier version of 19.35 1910. (Contributed by NM, 20-Sep-2003.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)

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		pm5.5 364	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
2	1	ralrexbid 3120	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜓))
3	2	biimpcd 252	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
4		rexnal 3115	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
5		pm2.21 124	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	reximi 3101	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
7	4, 6	sylbir 238	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
8		ax-1 6	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
9	8	reximi 3101	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
10	7, 9	ja 188	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
11	3, 10	impbii 212	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wral 3077  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3078  df-rex 3088

This theorem is used by:  r19.43  3131  r19.37v  3189  r19.36v  3191  r19.37  3266  r19.37zv  4463  r19.36zv  4468  iinexg  5309  bndndx  12605  nmobndseqi  31381  nmobndseqiALT  31382  unielss  44219  r19.36vf  46150