Theorem r19.35 3120

Description: Restricted quantifier version of 19.35 1897. (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		pm5.5 363	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓))
2	1	ralrexbid 3119	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜓))
3	2	biimpcd 251	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
4		rexnal 3114	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 𝜑)
5		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	reximi 3100	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
7	4, 6	sylbir 237	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
8		ax-1 6	. . . 4 ⊢ (𝜓 → (𝜑 → 𝜓))
9	8	reximi 3100	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
10	7, 9	ja 187	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓))
11	3, 10	impbii 211	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wral 3076  ∃wrex 3086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-ral 3077  df-rex 3087

This theorem is referenced by:  r19.43  3130  r19.37v  3188  r19.36v  3190  r19.37  3265  r19.37zv  4461  r19.36zv  4466  iinexg  5304  bndndx  12480  nmobndseqi  30982  nmobndseqiALT  30983  unielss  43795  r19.36vf  45714