Theorem rexanali 3085

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Proof shortened by Wolf Lammen, 27-Dec-2019.)

Assertion

Ref	Expression
rexanali	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

Proof of Theorem rexanali

Step	Hyp	Ref	Expression
1		dfrex2 3057	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
2		iman 401	. . 3 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
3	2	ralbii 3076	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
4	1, 3	xchbinxr 335	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀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-ex 1780  df-ral 3046  df-rex 3055

This theorem is referenced by:  nrexralim  3118  ceqsralbv  3626  frpoind  6318  frind  9710  qsqueeze  13168  ncoprmgcdne1b  16627  elcls  22967  ist1-2  23241  haust1  23246  t1sep  23264  bwth  23304  1stccnp  23356  filufint  23814  fclscf  23919  pmltpc  25358  ovolgelb  25388  itg2seq  25650  radcnvlt1  26334  pntlem3  27527  nosupbnd1lem5  27631  noinfbnd1lem5  27646  onscutlt  28172  umgr2edg1  29145  umgr2edgneu  29148  archiabl  33159  ordtconnlem1  33921  limsucncmpi  36440  matunitlindflem1  37617  ftc1anclem5  37698  clsk3nimkb  44036