Theorem r19.43 3093

Description: Restricted quantifier version of 19.43 1810. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.43	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.43

Step	Hyp	Ref	Expression
1		r19.35 3084	. 2 ⊢ (∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		df-or 385	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
3	2	rexbii 3041	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
4		df-or 385	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5		ralnex 2992	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
6	5	imbi1i 339	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
7	4, 6	bitr4i 267	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
8	1, 3, 7	3bitr4i 292	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383  ∀wral 2912  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  r19.44v  3094  r19.45v  3095  r19.45zv  4068  r19.44zv  4069  iunun  4604  wemapsolem  8455  pythagtriplem2  15522  pythagtrip  15539  dcubic  24573  legtrid  25486  axcontlem4  25847  erdszelem11  31183  soseq  31751  seglelin  32223  diophun  37337  rexzrexnn0  37368  ldepslinc  42298