Theorem r19.43 3102

Description: Restricted quantifier version of 19.43 1882. (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 3089	. 2 ⊢ (∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		df-or 848	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
3	2	rexbii 3077	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
4		df-or 848	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
5		ralnex 3056	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
6	5	imbi1i 349	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
7	4, 6	bitr4i 278	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
8	1, 3, 7	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847  ∀wral 3045  ∃wrex 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-ral 3046  df-rex 3055

This theorem is referenced by:  3r19.43  3103  r19.45v  3172  r19.44v  3173  r19.45zv  4469  r19.44zv  4470  iunun  5060  soseq  8141  wemapsolem  9510  pythagtriplem2  16795  pythagtrip  16812  dcubic  26763  addsdilem1  28061  mulsasslem2  28074  legtrid  28525  axcontlem4  28901  erdszelem11  35195  satfvsucsuc  35359  fmla1  35381  seglelin  36111  hashnexinjle  42124  fimgmcyclem  42528  rexor  42663  diophun  42768  rexzrexnn0  42799  dfvopnbgr2  47857  dfsclnbgr6  47862  ldepslinc  48502