Theorem rexanali 2998

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 2996	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
2		iman 440	. . 3 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
3	2	ralbii 2980	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
4	1, 3	xchbinxr 325	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀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-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  nrexralim  2999  wfi  5713  qsqueeze  12032  ncoprmgcdne1b  15363  elcls  20877  ist1-2  21151  haust1  21156  t1sep  21174  bwth  21213  1stccnp  21265  filufint  21724  fclscf  21829  pmltpc  23219  ovolgelb  23248  itg2seq  23509  radcnvlt1  24172  pntlem3  25298  umgr2edg1  26103  umgr2edgneu  26106  archiabl  29752  ordtconnlem1  29970  ceqsralv2  31607  frind  31740  nosupbnd1lem5  31858  limsucncmpi  32444  matunitlindflem1  33405  ftc1anclem5  33489  clsk3nimkb  38338