Theorem rexanali 3120

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 3093	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
2		iman 407	. . 3 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
3	2	ralbii 3112	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
4	1, 3	xchbinxr 338	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀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-ex 1813  df-ral 3081  df-rex 3091

This theorem is used by:  nrexralim  3150  ceqsralbv  3614  frpoind  6371  frind  9791  qsqueeze  13371  ncoprmgcdne1b  16869  matunitlindflem1  23049  elcls  23446  ist1-2  23720  haust1  23725  t1sep  23743  bwth  23783  1stccnp  23836  filufint  24294  fclscf  24399  pmltpc  25826  ovolgelb  25856  itg2seq  26118  radcnvlt1  26802  pntlem3  27993  nosupbnd1lem5  28126  noinfbnd1lem5  28141  oncutlt  28707  umgr2edg1  29849  umgr2edgneu  29852  archiabl  33858  extdgfialglem1  34435  ordtconnlem1  34667  limsucncmpi  37448  ftc1anclem5  38883  clsk3nimkb  45301