Theorem r19.43 3134

Description: Restricted quantifier version of 19.43 1917. (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 3124	. 2 ⊢ (∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		df-or 862	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
3	2	rexbii 3113	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
4		df-or 862	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5		ralnex 3092	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
6	5	imbi1i 352	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
7	4, 6	bitr4i 281	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
8	1, 3, 7	3bitr4i 306	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861  ∀wral 3080  ∃wrex 3090

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-or 862  df-ex 1813  df-ral 3081  df-rex 3091

This theorem is used by:  3r19.43  3135  r19.45v  3200  r19.44v  3201  r19.45zv  4467  r19.44zv  4468  iunun  5058  soseq  8213  wemapsolem  9581  pythagtriplem2  17039  pythagtrip  17056  dcubic  27231  addsdilem1  28594  mulsasslem2  28607  legtrid  29111  axcontlem4  29602  erdszelem11  36174  satfvsucsuc  36338  fmla1  36360  seglelin  37096  hashnexinjle  43445  fimgmcyclem  43863  rexor  43945  diophun  44050  rexzrexnn0  44077  nprmmul3  48884  dfvopnbgr2  49224  dfsclnbgr6  49229  ldepslinc  49952